Behavior of solutions and chaos in nonlinear reaction-diffusion PDE's related to cancer
aa r X i v : . [ m a t h . A P ] F e b BEHAVIOR OF SOLUTIONS AND CHAOS IN NONLINEARREACTION-DIFFUSION PDE’S RELATED TO CANCER
KAMAL N. SOLTANOV
Abstract.
In this paper, we study the mixed problem for new class of non-linear reaction-diffusion PDEs with the nonlocal nonlinearity with variable ex-ponents. Here we obtain results on solvability and behavior of solutions bothwhen these are yet dissipative and when these get become non-dissipative prob-lems. These problems are possess special properties: these can be to remaindissipative all time or can get become non dissipative after finite time. It isshown that if the studied problems get become non-dissipative can have aninfinite number of different unstable solutions with varying speeds and also aninfinite number of different states of spatio-temporal (diffusion) chaos that aregenerated by cascades of bifurcations governed by the corresponding steadystates. The behavior of these solutions is analyzed in detail and it is explainedhow space-time chaos can arise. Since these problems desctibe the dynamicsof cancer, we explain the behavior of the dynamics of cancer using obtainedhere results. Introduction
In this article we study the mixed problem for new class of nonlinear reaction-diffusion equations with variable nonlinearity that are the parabolic partial differ-ential equations (PDE’s). The studied here equations possess special properties andare a mathematical models that describe of the dynamics of cancer. The theory ofnonlinear PDE’s of parabolic type is of great interest both in itself and also as theuseful mathematical model for a wide variety of important problems. This theory iswidely used under studies various problems of hydrodynamics, theory of nonlineardiffusion and also under studies many problems in physics, chemistry, biology etc.For further information on applications, the reader is referred to such sources as[1, 4, 9, 10, 11, 15, 16, 22, 27, 27, 28, 29, 35, 39], etc..Many biological processes at mathematical modeling are describe by the non-linear reaction-diffusion equations (reaction-diffusion-convection or advection). Inaddition, the mathematical modelling of the dynamics of the process of diseasesdue of infection be described by the reaction-diffusion equation also. It should benoted that for study of the dynamics of cancer also are used the nonlinear reaction-diffusion equations ([2, 3, 7, 8, 13, 14, 19, 23, 25, 26, 31, 32, 34, 38]).It is need to note the nonlinear reaction-diffusion PDEs presentes also greatinterest in a purely mathematical sense, as these comprise an infinite-dimensionaldynamical systems that exhibit an more spectrum of solution phenomena. The
Mathematics Subject Classification.
Primary 35J10, 35K05, 35R30; Secondary 37G30,37B25, 34C23, 92C15;
Key words and phrases.
Semilinear PDE, reaction-diffusion equation, behavior of solutions,chaos, dynamics of cancer. nonlinear parabolic PDE’s comprise an infinite-dimensional dynamical systems thatexhibit an amazing spectrum of solution phenomena including traveling waves,dissipative solitons, spiral waves, target patterns, bifurcation cascades, chaos andlong-time dynamical configurations of great complexity ([ 15, 16, 17, 22, 24, 27, 28,29, 30, 32, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44 , 45, 46, 47, 48, 49, 50]). Especiallyinterest is investigation of the long-time behavior of the solutions, which concernsthe dynamical regimes that the system may settle into as the time greate. If thesystem is dissipative, there is typically a global strange attractor, but things areespecially challenging mathematically when the system is non-dissipative inasmuchas the long-time dynamics can be much more diverse and complicated than in thedissipative case. It is need to note that long-time dynamics of solutions of nonlinearreaction-diffusion equations in non-dissipative case have many open problems. Inthis article we obtain new results on the finite and long-time dynamics of classesof nonlinear reaction-diffusion equations, used for mathematical models of manyinteresting phenomena, which provide additional mathematical details regardingblow-up and chaotic transitions for systems analyzing in the literature [1, 4, 5, 11,12, 24, 30, 37, 40, 44 , 47, 49, 50]. Moreover, we study the long-time dynamicsof classes of nonlinear reaction-diffusion equations which during of time pass fromdissipative to non-dissipative equations.Many authors for study of the dynamics of propagation of cancer account that themathematical model describing this process is a reaction-diffusion PDE’s. More-over, is assumed that this model feel of the changes of the state of the cells ac-cording to time. According medical investigations known that genes can either beactivated or suppressed, when signals stimulate receptors on the cell surface andare then transmitted to the nucleus of the cell. The reception of particular signalscan induce a cell to reproduce itself in the form of identical descendants, that is theso-called clonal expansion or mitosis, or to die, that is the so-called apoptosis orprogrammed death (e.g. [6, 7, 8, 13, 14, 19, 20, 23, 24, 25, 26, 31, 32, 33, 34, 38] andthe references therein). We use a reaction-diffusion PDE’s for study of the sameproblem also, but our approach is different from models were used in the above-mentioned works. Here we take account of the mentioned properties by selectingof the coefficients and exponents of nonlinearity as the functions which are depen-dents of variables ( t, x ). Since the changes of the state of the cells pass during of thetime, one need to use such model that can to feel of these changes. According tothis properties it is require take account that the changes of cells is the spatiotem-poral process that in particular, are arose in appear of the cancer. Consequently,studying the properties of the solutions of these PDEs helps explain the dynamicsof the natural processes under discussion in applications, for example, the resultsobtained here can be used to analyze the dynamics of cancer spread. In this paper,we want to explain this dynamics and how cell changes occur over a long-time.In what follows we study a mixed problem for an equation that changes from dis-sipative to non-dissipative reaction-diffusion PDEs with increasing time. Moreover,the corresponding steady-state problem passes from a uniquely solvable problem toa problem with an infinitely many solutions. It is shown that the trajectories ofsolutions in the phase space depend on the choice of starting point on a sphere ofthe initial values. Therefore the behavior of solutions to this problem can be morecomplicated. Here can be arise many variants, e.g. can be the blow-up at finitetime, can be exists such absorbing manifold, that the associated dynamics tends
EHAVIOR OF SOLUTIONS AND CHAOS 3 to be chaotic. The class of nonlinear reaction-diffusion equations studied here arerelevant to investigations of the cancer.In this article we investigate a reaction-diffusion problem that gradually is con-verted to the advection reaction-diffusion process with the spatiotemporal chaos.So, we will study the mixed problem for the following equation as first approach tothe posed question(1.1) ∂u∂t = ∇ · ( a ( x ) ∇ u ) − b ( x ) u − f ( t, x ) k u k p ( t ) u + g ( t, x ) k u k p ( t ) u, ( t, x ) ∈ R + × Ωwhere Ω ⊂ R n , n ≥ ∂ Ω, u ( t, x ) is unknown function and condition ( i ) a ( x ) , b ( x ) , f ( t, x ) , g ( t, x ) > , p ( t, k u k ) ≥ , p ( t, k u k ) ≥ k u k denote the density of all cells from the domain Ω and p ( t ) denote change of the range of normal cells, and p ( t ) denote change of allcells unreceiving signals from immune system in Ω during of time t . In the aboveequation assumed the function g ( t, x ) > t, x ). The cellular scale refers to the main (interactive)activities of the cells: activation and proliferation of tumor cells and competitionwith immune cells. Since the cells from one state to another pass gradually, i.e.from immune state to the activation and proliferation (interactive) state. The(interactiveness) activities of the cells after infections mainly are changed in thefollowing sequence: activation and proliferation of tumor cells and competition withimmune cells. As one can see in the works dedicated to the cancer the differentstates of cells usually denoted by different notations (see, e.g. mentioned above),but the offered here mathematical model feel of these different states by coeffitients KAMAL N. SOLTANOV and exponents. In the equation (1.1) a ( x ) is the coefficient of diffusion measuringthe mobility of any cells (namely, the immune cells and proliferating cells) and thefunctions f ( t, x ) and g ( t, x ) are the recruitment rate of susceptible and degeneratedcells, respectively.So, we will study the following homogeneous problem: the equation (1.1) withthe initial and boundary conditions(1.2) u (0 , x ) = u ( x ) , u ( t, x ) ≥ , (1.3) u ( t, x ) (cid:12)(cid:12) ∂ Ω × R + = 0 , p ( t ) ≥ , p (0) = p − . In this problem assumed that the external interferences are excluded.Let the following conditions ( ii ) p ( t ) ≥ ∃ t > , t > t = ⇒ p ( t ) > p ( t ) ; ∃ t > t , t > t = ⇒ p ( t ) = 0;hold, where t defined by equality t = inf n t ∈ R + (cid:12)(cid:12)(cid:12) p ( t ) p ( t ) > o . Where, ingeneral, p ( t, τ ) ց and p ( t, τ ) ր , g ( t, x ) ր if t, τ ր , b ( x ) > g (0 , x ) ≥
0. Here p > p and p are studied. In Section 3 the long-time dynamics ofthe solutions of the exmined problem is investigated that has 2 subsections wherethe blow-up under finite time and the long-time behavior of solutions, and alsochaotics are studied.2. Solvability of problem (1.1)-(1.3)
Let Ω ⊂ R n , n ≥
1, is bounded domain with sufficiently smooth boundary ∂ Ω(at least from the Lipschitz class). By W , (Ω) ≡ H (Ω) we denote the Sobolevspace and by L r (Ω), r ≥ i ) and arebounded continuous functions, where Q T = (0 , T ) × Ω, T > ∇ · ( a ∇ ) : H (Ω) −→ H − (Ω). EHAVIOR OF SOLUTIONS AND CHAOS 5
Now we cosider the solvability of this problem, which will be analyzed makinguse of the general results from [41]. We take u ∈ B H r (0) , where r > , andstudy the operator A generated by the problem (1.1)-(1.3): it acts, by definition,from X := W , (cid:0) , T ; H − (Ω) (cid:1) ∩ L (cid:0) , T ; H (Ω) (cid:1) ∩ { u ( t, x ) | u (0 , x ) = u } to Y = L (cid:0) , T ; H − (Ω) (cid:1) . Consequently, as the solution of the problem (1.1)-(1.3)we understand follows: a function u ∈ X called the solution of this problem if itsatisfies this problem in the sense the space Y = L (cid:0) , T ; H − (Ω) (cid:1) .According to conditions ( i ) and ( ii ) the equation (1.1) can changes its characterfrom the dissipative equation to the non-dissipative, whence ensue that here is pos-sible the following different cases, which necessary investigate separately: namely,there exist such times 0 ≤ t ≤ t < ∞ that (1) p ( t ) > p ( t ), t ∈ [0 , t ]; (2) p ( t ) ≥ p ( t ) for t ∈ [ t , t ]; (3) p ( t ) = 0 for t > t . Since the character of theproblem (1.1)- (1.3) also will changes depending on the analysed case.2.1. Existence in the case p ( t ) > p ( t ) . So, let p ( t ) > p ( t ). In the beginningit is needed to obtain a priori estimates for the possible solutions, therefore weconsider the following expression (cid:28) ∂u∂t , u (cid:29) = D ∇ · ( a ∇ u ) − b u − f k u k p ( t ) u + g k u k p ( t ) u, u E = ⇒ and in sequal having carried out some calculations we get12 ddt k u k = − h a ∇ u, ∇ u i + (cid:28) a ∂u∂ν , u (cid:29) | ∂ Ω − h b u, u i − k u k p ( t ) h f u, u i ++ k u k p ( t ) h g u, u i = − (cid:13)(cid:13)(cid:13) a ∇ u (cid:13)(cid:13)(cid:13) + (cid:28) a ∂u∂ν , u (cid:29) | ∂ Ω −− (cid:13)(cid:13)(cid:13) b u (cid:13)(cid:13)(cid:13) − k u k p ( t ) (cid:13)(cid:13)(cid:13) f ( t ) u (cid:13)(cid:13)(cid:13) + k u k p ( t ) (cid:13)(cid:13)(cid:13) ( g ) ( t ) u (cid:13)(cid:13)(cid:13) according of the boundary condition12 ddt k u k = − (cid:13)(cid:13)(cid:13) a ∇ u (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) b u (cid:13)(cid:13)(cid:13) − k u k p ( t ) (cid:13)(cid:13)(cid:13) f u (cid:13)(cid:13)(cid:13) +(2.1) + k u k p ( t ) (cid:13)(cid:13)(cid:13) g u (cid:13)(cid:13)(cid:13) , k u k (0) = k u k where h◦ , ◦i = R Ω ◦ × ◦ dx , k◦k is the norm of L (Ω) = H (Ω).For estimate of the last term in the equation (2.1) we consider the expression g ( t, x ) · k u k p ( t ) due of k u k p ( t ) (cid:13)(cid:13)(cid:13) g u (cid:13)(cid:13)(cid:13) = Z Ω g u k u k p ( t ) dx = ⇒ g u k u k p ( t ) . Hence use ε − Young inequality we have g k u k p ( t ) ≤ p − p p (cid:16) ε p p − p g p p − p f − p p − p (cid:17) + p p (cid:16) ε − p p f k u k p ( t ) (cid:17) , Where B H r (0) = (cid:8) u ∈ H | k∇ u k ≤ r (cid:9) , need to note k∇ u k ≡ k u k H . We wish to note when degradation of cells begin p ( t ) remains less than p ( t ) until sometime t . By conditions take place: p ( t ) ր and p ( t ) ց when t ր . KAMAL N. SOLTANOV and taking ε = (cid:16) p p (cid:17) p p (2.2) g k u k p ( t ) ≤ f k u k p ( t ) + "(cid:18) p p (cid:19) p p − p − (cid:18) p p (cid:19) p p − p gf (cid:19) p p − p . If denote by b ( t, x ) the following expression b ( t, x ) = b ( x ) − (cid:18) p ( t ) p ( t ) (cid:19) p t ) p t ) − p t ) − (cid:18) p ( t ) p ( t ) (cid:19) p t ) p t ) − p t ) (cid:18) g p ( t ) ( t, x ) f p ( t ) ( t, x ) (cid:19) p t ) − p t ) then from (2.1) we derive the inequality(2.3) 12 ddt k u k ≤ − Z Ω a ( x ) |∇ u | dx − Z Ω b ( t, x ) u dx Hence one need to examine 2 variants: (a) b ( t, x ) ≥ b ( t, x ) <
0. Lettakes place (a) then inequation(2.4) Z Ω a ( x ) |∇ u | dx + Z Ω b ( t, x ) u dx ≥ . holds and we derive the following problem12 ddt k u k ≤ − Z Ω a ( x ) |∇ u | dx − Z Ω b ( t, x ) u dx, k u k (0) = k u k . Assume there exist such numbers a , A , b , B > a ≤ a ( x ) ≤ A and b ≤ b ( x ) ≤ B . Whence if we denote by b ( t ) = inf { b ( t, x ) | x ∈ Ω } then we getthe problem 12 ddt k u k ≤ − a k∇ u k − b ( t ) k u k , k u k (0) = k u k . Then one can affirm that solutions will be bounded. Moreover, maybe a solutionwill remained stable when t ր ∞ if p ( t ) will remains less than p ( t ) for any t > Now let takes place (b) then if we denote by e b ( t ) = sup { b ( t, x ) | x ∈ Ω } thenwe have the problem12 ddt k u k ≤ − a k∇ u k + e b ( t ) k u k , k u k (0) = k u k . In this case if a k∇ u k ≥ (cid:12)(cid:12)(cid:12)e b ( t ) (cid:12)(cid:12)(cid:12) k u k then the previous assertion occurs.Since the case of nonfufilment of (2.4) is particular case of the studied in thesubsection 3, here we not will discuss on this.Thus the following result is proved. Remark 1.
Consequently, the investigated process either isn’t make worsen or will improve andremains in the bounded of vicinity of zero when t ր ∞ . EHAVIOR OF SOLUTIONS AND CHAOS 7
Theorem 1.
Let p ( t ) < p ( t ) for t ∈ [0 , t ) . Let all above conditions and (2.4)are fulfilled.Then the problem (1.1)-(1.3) solvable and remains in the bounded of vicinity ofzero for t ∈ [0 , t ) . From this theorem and inequalities of such type as (2.2) and (2.4) follows
Corollary 1.
If is fulfilled case (1), i.e. p ( t ) < p ( t ) and g ( t, x ) < f ( t, x ) for t ∈ [0 , t ) then the problem (1.1)-(1.3) solvable and remains in the bounded ofvicinity of zero for t ∈ [0 , t ) . The solvability of the problem for t ∈ [0 , t ) in this case follows from generalresults of [40, 41, 44 ] by virtue of equation (2.1). Remark 2.
It is clear that the immune system of organism act to ”infected” cellsin order to stop of propagation of the degeneration or reanimate of such cells. If theimmune system is sufficiently strong then it can reanimate of certain parts degen-erated cells or, at least, stop of the continiotion of the degeneration. Consequently,if this is possible then the exponent p ( t ) can vary and maybe, not will increase. Analysis of the case p ( t ) = p ( t ) . According to conditions p ( t ) ց and p ( t ) ր when t ր , because there exists such time t > p ( t ) = p ( t ). Inthis case the relation between f ( t, x ) and g ( t, x ) can changed depending of pointsof Ω.It is clear that if p ( t ) = p ( t ) and g ( t, x ) ≤ f ( t, x ) for t > t then the propa-gation of the possible solution will rightly determinable as in previous subsection,and the case when g ( t, x ) > f ( t, x ) for t > t will be to study later as the case p ( t ) ≥ p ( t ) for t ≥ t . But if the relation between g ( t, x ) and f ( t, x ) for t > t is undetermined then maybe arise a chaos. Therefore, will be best if this case toinvestigate together of the case p ( t ) ≥ p ( t ).2.3. Solvability in the case p ( t ) > p ( t ) . Let the case (3) is fulfilled, i.e. p ( t ) ≥ p ( t ) for t ≥ t . If p ( t ) > p ( t ) then independent of the relation be-tween g ( t, x ) and f ( t, x ) the behavior of the possible solutions generally speakingwill indeterminable, as their behavior will has vary according of p ( t ), of the initialvalue and the spectrum of the Laplace operator. It should be noted that accordingto conditions f ( t, x ) , p ( t ) ց and g ( t, x ) , p ( t ) ր as t ր , therefore acros sometime t > t will be: g ( t, x ) > f ( t, x ) and p ( t ) ≫ p ( t ).So, let us p ( t ) > p ( t ) for t ≥ t and again will examine (2.1), more exactly12 ddt k u k + Z Ω h a ( x ) |∇ u | + b ( x ) u + k u k p ( t ) f ( t, x ) u i dx =(2.5) = k u k p ( t ) Z Ω g ( t, x ) u dx, k u k ( t ) = k u t k . The expression k u k p ( t ) f ( t, x ) one can estimate by use of the expression k u k p ( t ) g ( t, x ) like in subsection 2.1(2.6) f k u k p ( t ) ≤ p − p p (cid:18) f p g p (cid:19) p − p + p p g k u k p ( t ) . Probably, in this case such constant always will be exists
KAMAL N. SOLTANOV
Consequently, there exist such functions b ( t, x ) and g ( t, x ) that the followinginequality12 ddt k u k + Z Ω h a ( x ) |∇ u | + b ( t, x ) u i dx − k u k p ( t ) Z Ω g ( t, x ) u dx ≤ b ( t, x ) = b ( b, g, f, p , p ) ; g ( t, x ) = g ( g, p , p ) . Thus, from (2.5) we derive12 ddt k u k + Z Ω h a ( x ) |∇ u | + b ( t, x ) u i dx − (2.7) − k u k p ( t ) Z Ω g ( t, x ) u dx ≤ , k u k (0) = k u k . So, from the Cauchy problem (2.5) we derive the Cauchy problem (2.7) for the in-equation, which permit we to investigate of the behavior of solutions of the problem(2.5).Denote by g = sup { g ( t, x ) | x ∈ Ω } and b = inf {| b ( t, x ) | | x ∈ Ω } whichwhere exist due of conditions of this problem. We assume that the Laplace operator − ∆ : H (Ω) −→ H − (Ω) has only a point spectrum, i.e. σ ( − ∆) = σ p ( − ∆) ⊂ R + and denote by λ the minimal eigenvalue of the Laplace operator − ∆ (we will notethat in this case k∇ u k ≥ λ k u k be valid).We now investigate the problem (2.7) for the initial data satisfying the condition u ∈ S H r (0) ⊂ H (Ω) if g (cid:16) e − ( a λ + b ) t r (cid:17) p < a λ + b ( t ) , where b = sup { b ( t ) | t ∈ (0 , t ) } . Then from ( ?? ) we get12 ddt k u k ≤ − a k∇ u k − b k u k + g k u k p ( t )+2 ≤− (cid:16) a λ + b − g k u k p ( t ) (cid:17) k u k , k u ( t ) k | t = t = k u ( t ) k . Thus we have k u k ≤ exp {− ( a λ + b − g r p ) t } k u ( t ) k . Consequently, one can formulate the following solvability result for the problem(1.1)-(1.3)
Theorem 2.
Let all above conditions on the problem (1.1)-(1.3) are fulfilled. Thenthis problem solvable for any u ∈ B H r (0) ⊂ H (Ω) if r satisfies the inequality g (cid:16) e − ( a λ + b ) t r (cid:17) p < a λ + b ( t ) . Moreover the mapping (semi-flow) S ( t ) : u t −→ u ( t ) is such that H strongly S : (cid:16) B H r (0) (cid:17) ց as t ր ∞ . Where S H r (0) = (cid:8) u ∈ H | k∇ u k = r (cid:9) . EHAVIOR OF SOLUTIONS AND CHAOS 9 Longtime behavior of solutions
Now we will study the case when p ( t ) = 0 for t ≥ t . Moreover, since theinequality (2.6) shows that if p ( t ) > p ( t ) then always instead of the equation from(2.5) one can use the inequation (2.7), which is sufficiently for the investigation ofthe behavior of solutions. One can make use of the formal solution to the problem(1.1)-(1.3) by takes account of the above assumption onto the known functions.But if the known functions depend at both of variable then the study of the posedproblem become very complicate. Therefore, we will investigate the behavior ofsolutions of the problem in somewhat weak form. As will be to seen that suchapproach is enough to receive the necessary information about the behavior ofsolutions.3.1. Blow-up Solutions.
So, we will begin to investigate the behavior of solutionsof posed problem when p ( t ) = 0.Let us p ( t ) = 0 for t ≥ t . We will examine again (2.1) written in the followingform 12 ddt k u k + Z Ω h a ( x ) |∇ u | + ( b ( x ) + f ( t, x )) u i dx − (3.1) − k u k p ( t ) Z Ω g ( t, x ) u dx = 0 , k u k ( t ) = k u t k . According to conditions there exist such constants a , A , b , B , f , F , g , G > a ≤ a ( x ) ≤ A ; b ≤ b ( x ) ≤ B ; f ≤ f ( t, x ) ≤ F ; g ≤ g ( t, x ) ≤ G hold.Consequently, we will study the following problems(3.2) 12 ddt k u k + A k∇ u k + ( B + F ) k u k − g k u k p ( t )+2 ≥ ddt k u k + a k∇ u k + ( b + f ) k u k − G k u k p ( t )+2 ≤ k u k ( t ) = k u t k , which can presents the behavior of thepossible solutions of the problem (3.1). More exactly, the trajectory of the possiblesolutions of the problem (3.1) in phase space go between of the boundary layersdescibed by solutions of the above problems.So, from problem (3.2) and (3.3) we derive the following problems(3.4) 12 ddt k u k + A k∇ u k + B k u k − g k u k p ( t )+2 ≥ ddt k u k + a k∇ u k + b k u k − G k u k p ( t )+2 ≤ , with the initial condition k u k ( t ) = k u t k , where B = B + F and b = b + f . Remark 3.
The above inequalities show that it is necessary used the derivative ofthe function with the variable exponent, therefore here we deduce it. Consider thefunction y ( t ) − q ( t ) then ddt (cid:16) y ( t ) − q ( t ) (cid:17) = ddt exp {− q ( t ) ln y ( t ) } = − exp {− q ( t ) ln y ( t ) } ddt ( q ( t ) ln y ( t )) == − exp {− q ( t ) ln y ( t ) } (cid:20) q ´ ( t ) ln y ( t ) + q ( t ) y ( t ) y ´ ( t ) (cid:21) =(3.6) = y ( t ) − q ( t ) (cid:20) q ´ ( t ) ln y ( t ) + q ( t ) y ( t ) y ´ ( t ) (cid:21) . Consequently, for investigate of the inequality (3.5) we will rewrite it in thefollowing form12 ddt k u k + k u k (cid:18) p ´( t )2 p ( t ) ln k u k (cid:19) ≤ − a k∇ u k − b k u k +(3.7) + G k u k p ( t )+2 + k u k (cid:18) p ´( t )2 p ( t ) ln k u k (cid:19) . Remark 4.
Consider the function h ( z ) = z − ln z for z > , then we have ddt h ( z ) = z − − z = z (cid:16) z − (cid:17) . Whence for ddt h ( z ) = 0 we get z − or z = 4 is the minimum of function h ( z ) , i.e. min { h ( z ) | z > } = 2 − ln 4 > .Consequently, h ( z ) > for z > (see, e.g. the proof of Lemma 9 of [45] ). Thus we get(3.8) k u k (cid:18) p ´( t )2 p ( t ) ln k u k (cid:19) ≤ k u k ≤ ( b + f ) k u k + c k u k p ( t )+2 as p ( t ) ≥
1, where c = c ( b , f , p ). More exactly, use the ε − Young inequalityand selecting ε = p ´1 p (cid:20)(cid:18) p p − (cid:19) ( b + f ) (cid:21) − p − p = p ´1 p (cid:20)(cid:18) p p − (cid:19) b (cid:21) − p − p we obtain c = ε p p .Using the inequality (3.8) in (3.7) we derive the following problem(3.9) 12 ddt k u k + k u k (cid:18) p ´( t )2 p ( t ) ln k u k (cid:19) ≤ − a k∇ u k + ( G + c ) k u k p ( t )+2 with the initial condition k u k ( t ) = S ( t ) k u k =(3.10) = exp − a λ t + t Z b ( s ) ds + t Z t (cid:16) b ( s ) − g ( s ) r p ( s )0 (cid:17) ds k u k where u ∈ H is a given initial function, e.g. one can to choose it from theball B H e r (0) ⊂ H with the radius e r >
0. So, we study this problem for theinitial function u belonging to the B H e r (0). In the previous section was proved EHAVIOR OF SOLUTIONS AND CHAOS 11 that problem has solution in appropriate space for t ∈ (0 , t ], consequently we canaffirm the value of k u k ( t ) is defined on t as in (3.10) with the semi-flow S ( t ).Thus if one denote k u k ( t ) = y ( t ), p ( t ) = p ( t )2 and account k∇ u k ≥ λ k u k then (3.9)-(3.10) we derive the following Cauchy problem12 dydt + y (cid:18) p ´( t ) p ( t ) ln y (cid:19) ≤ − a λ y + G y p ( t )+1 , y ( t ) = y t . here is denoted p ( t ) = p ( t )2 . Whence follows p ( t ) y − p ( t ) − dydt + y − p ( t ) ( p ´( t ) ln y ) ≤ − a λ p ( t ) y − p ( t ) + G p ( t )and consequently, we have the inequality − d (cid:0) y − p ( t ) (cid:1) dt ≤ − a λ p ( t ) y − p ( t ) + G p ( t )denoted by z ( t ) = y − p ( t ) we get dzdt ≥ a λ p ( t ) z − G p ( t ) , z ( t ) = y ( t ) − p ( t ) . Hence follows z ( t ) ≥ e a λ t R t p ( s ) ds z ( t ) − t Z t e a λ t R τ p ( s ) ds G p ( τ ) dτ . Assume ddt P ( t ) = p ( t ) then we obtain z ( t ) ≥ e a λ ( P ( t ) − P ( t )) z ( t ) + G t Z t e a λ ( P ( t ) − P ( τ )) p ( τ ) dτ G Consequently, we arrive z ( t ) ≥ e a λ ( P ( t ) − P ( t )) z ( t ) + h − e a λ ( P ( t ) − P ( t )) i G a λ == G a λ − e a λ ( P ( t ) − P ( t )) (cid:20) G a λ − z ( t ) (cid:21) . Takes account that z ( t ) = y − p ( t ) we get y − p ( t ) ≥ G a λ − e a λ ( P ( t ) − P ( t )) (cid:20) G a λ − y ( t ) − p ( t ) (cid:21) or y p ( t ) ≤ e − a λ ( P ( t ) − P ( t )) y ( t ) p ( t ) n − (cid:2) − e − a λ ( P ( t ) − P ( t )) (cid:3) G y ( t ) p ( t a λ o == e − a λ ( P ( t ) − P ( t )) a λ y ( t ) p ( t ) n a λ − (cid:2) − e − a λ ( P ( t ) − P ( t )) (cid:3) G y ( t ) p ( t ) o . Therefore one need to investigate the following equation a λ − G y ( t ) p ( t ) + e − a λ ( P ( t ) − P ( t )) G y ( t ) p ( t ) = 0 or P ( t ) = P ( t ) − a λ ln − a λ G y ( t ) p ( t ) ! since P ( t ) is the increasing function its inverse function ( P ( t )) − exists thereforewe can derive the upper bound of the blow-up time.If one lead the above-mentioned calculations for the Cauchy problem posed forinequality (3.4) then we derive the following inequation y p ( t ) ≥ e − ( A λ + e B ) ( P ( t ) − P ( t )) (cid:16) A λ + e B (cid:17) y ( t ) p ( t ) n(cid:16) A λ + e B (cid:17) − h − e − ( A λ + e B ) ( P ( t ) − P ( t )) i G y ( t ) p ( t ) o . Consequently, we arrive to result similar of the obtained above result. Whence bysimilar way as of the previous case we can determine the lower bound of the blow-up time. Therefore blow-up time t col of the main problem can be found between ofthese times that is finite since p is finite number, and p ( t ) is continuous functionand satisfies condition p ( t ) ≤ p − Theorem 3.
Let us the initial function u ∈ H is such that the condition (cid:13)(cid:13)(cid:13) a ∇ u ( t ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) g ( t ) u ( t ) (cid:13)(cid:13)(cid:13) k u ( t ) k p ( t ) < is fulfilled. Then each solution u ∈ X of the problem (1.1) - (1.3) has blow-up in H a finite time. Notation 1.
Whence one can derive condition on u (0) using of the adduced abovedefinition of u ( t ) . Remark 5.
The result of this theorem shows that if is fulfilled the conditions p ( t ) ≪ p ( t ) or p ( t ) = 0 then almost everywhere in the domain Ω were unre-mained of the immune cells and depending on the initial condition may be collapse. Solution Behavior.
Consider the equation (1.1) in the case when p ( t ) = 0 ∂u∂t = ∇ · ( a ( x ) ∇ u ) − [ b ( x ) + f ( t, x )] u + g ( t, x ) k u k p ( t ) u. For study of the behavior of solutions we will investigate the following problem(3.11) ∂u∂t = b a ∆ u − hb b + b f ( t ) i u + b g ( t ) k u k p ( t ) u, u ( t ) = u t , where instead of the functions a, b, f and g set their mean values that denoted by b a, b b, b f and b g , e.g. b a = mes Ω R Ω a ( x ) dx , in addition we denote b b + b f ( t ) by b b ( t ).It should be noted according of the results of [ 18] on the differential operatorsof elliptic type, study of the behavior of solutions of the problem for the equation(3.11) will give the sufficient informations about the behavior of solutions and alsofor the main problem. Consequently, if we wish obtain such result as in the above section then necessary to selectthe initial function by appropriate way as in above section.
EHAVIOR OF SOLUTIONS AND CHAOS 13
So, for study of the behavior of solutions of the problem (3.11) consider thefollowing problem(3.12) 12 ddt k u k + b a k∇ u k + b b ( t ) k u k − b g ( t ) k u k p ( t )+2 = 0 , (3.13) k u ( t ) k = k u t k , where k u ( t ) k = S ( t ) k u k the semi-flow S ( t ) defined in (3.10).Thus we arrive to the problem that is similar with the problem studied in ourarticle [44 ]. In the other words we can lead analogously reasoning as in the citedarticle in order to study of the behavior of solutions of the problem (3.12)-(3.13),since this problem is similar to first problem studied in the mentioned article.Let u t ∈ H (Ω) with the norm k◦k H (Ω) ≡ k∇◦k . We assume that the Laplaceoperator − ∆ : H (Ω) −→ H − (Ω) has only a point spectrum, i.e. σ ( − ∆) ≡ σ P ( − ∆) ⊂ (0 , ∞ ) . Denote the eigenvalues of the Laplace operator − ∆ : H (Ω) −→ H − (Ω) by λ j , j = 1 , , ... ( σ P ( − ∆) ≡ { λ j | j ∈ N } . This, of course, requires that the domain Ω is sufficiently regular in a geometricsense.Now we consider the problem (3.12) - (3.13) and investigate the inverse mappingof the operator b A generated of this problem. For simplicity, we assume that the eigenfunctions and adjoint eigenfunctions aretotal (complete) in the space H (Ω) ; moreover, we may assume without loss ofgenerality that they generate an orthogonal basis in this space.Let inf { λ j ∈ σ P ( − ∆) : λ j > r ρ , j = 1 , , ... } = λ k . Then, we can represent ([ 15, 20, 28, 30, 32, 40, 49]) the space H (Ω) in the form H (Ω) ≡ H k ⊕ H − k , where the subspace H k ⊂ H (Ω) is related to { λ j } k − j =1 andhas dimension dim H k = k − H − k is a subspace of codim H − k = k − . Wecan now introduce the projections Q k and P k ; Q k : H (Ω) −→ H − k ⊂ H (Ω)and P k : H (Ω) −→ H k ⊂ H (Ω) , giving rise to the splitting u ≡ Q k u + P k u. (It is well known that such a decomposition allows to introduce either a spectralmeasure or a family of spectral projections, see for example, [ 15, 19, 40, 48], etc.)Thus, it is easy to see that − ∆ : H k −→ H − k and − ∆ : H − k −→ H −− k , wherethe subspaces H − k , H −− k possess bi-orthogonal bases (see, for instance, [16, 37, 38,39], etc.), and due to the evident commutativity of operators P k and Q k with theLaplacian ∆ in H (Ω) , one can rewrite the problem as(3.14) ∂∂t P k u − b a ∆ P k u + b b ( t ) P k u − b g ( t ) k u k p ( t ) P k u = 0 , (3.15) P k u ( t , x ) = P k u t ( x ) ∈ H k ⊂ H (Ω) , (3.16) ∂∂t Q k u − b a ∆ Q k u + b b ( t ) Q k u − b g ( t ) k u k p ( t ) Q k u = 0 , (3.17) Q k u ( t , x ) = Q k u t ( x ) ∈ H − k ⊂ H (Ω) . Operator b A is defined as the operator A in Section 2. As our aim is the investigation of the behavior of solutions of the problem underthe condition u t ∈ B H (Ω) r ( t ) (0) then it is enough to assume that b aλ k − + b b ( t ) < b g ( t ) r ( t ) p ( t ) < b aλ k + b b ( t ). Then for the problem (3.12) - (3.13) we obtain0 = 12 ddt k P k u k ( t ) +(3.18) + hb a k∇ P k u k + b b ( t ) k P k u k − b g ( t ) k u k p ( t ) k P k u k i ( t ) , (3.19) h P k u, P k u i | t = t = k P k u k ( t ) = k P k u t k . Whence, it follows that for some t > t for t ∈ [ t , t ) we have b g ( t ) k u k p ( t ) ( t ) ≤ b g ( t ) r ( t ) p ( t ) + θ < b aλ k + b b ( t ) , for some θ >
0. Indeed if k u t k = r ( t ), then we have from (3.18) - (3.19) that ddt k P k u k ( t ) + 2( b aλ k − + b b ( t ) − r ( t ) p ( t ) b g ( t ) k P k u k ( t ) ≥ ddt k P k u k ( t ) + 2( b aλ k − + b b ( t ) − b g ( t ) r ( t ) p ( t ) ) k P k u k ( t ) ≥ k P k u k ( t ) ≥ exp − t Z t (cid:16)b aλ k − + b b ( s ) − b g ( s ) r p ( s )0 (cid:17) ds k P k u k . Thus, we see that if k Q k u k ( t ) ≤ δ < θ < r for some sufficiently small δ > t ∈ [ t , t ) , then the solution of problem (3.18) - (3.19) exists and is anexponentially increasing function.Now consider the problem (3.16) - (3.17) for which one easily obtains12 ddt k Q k u k ( t ) + b a k∇ Q k u k ( t ) + b b ( t ) k Q k u k − (3.21) − b g ( t ) (cid:16) k u k p ( t ) k Q k u k (cid:17) ( t ) = 0 , h Q k u, Q k u i | t = t = k Q k u k ( t ) = k Q k u t k . Therefore, the solution of problem (3.21) exists and is an exponentially decreasingfunction. Consequently for k P k u t k + k Q k u t k = r ( t ) if k Q k u t k < k P k u t k and k Q k u t k is sufficiently small, then the solution k u k ( t ) exists and is an increasingfunction up to some time.For a detailed study of the behavior of solutions to the problem we make useof the following assumption: the system of eigenfunctions { w k } ∞ k =1 ⊂ H (Ω) com-prises an orthonormal basis of this space. Then each function u ( t, x ) ∈ L (cid:0) (0 , T ) ; H (Ω) (cid:1) has the representation u ( t, x ) = ∞ P k =1 u k ( t ) w k ( x ) . Consequently, according to (3.12)- (3.13), the problem is equivalent to studying the system of equations12 ddt | u k ( t ) | + (cid:16)b aλ k + b b ( t ) (cid:17) | u k ( t ) | − EHAVIOR OF SOLUTIONS AND CHAOS 15 (3.22) − b g ( t ) ∞ X i =1 | u k ( t ) | ! p t )2 | u k ( t ) | = 0with the Cauchy data(3.23) | u k ( t ) | = | u t k | , k = 1 , , ..., k . Let u ( t , x ) ∈ B H (Ω) r ( t ) (0) and k u t k < r ( t ), then we have k u k ( t ) ≡ ∞ X i =1 | u i ( t ) | ! ≤ r ( t ) + ε for sufficiently small t = t ( ε, r ( t )) > t . But | u k ( t ) | increases in this case for k =1 , , ..., e k ≤ k depending on the relationship between k u t k p ( t ) and λ k (therefore,between r ( t ) and λ k ).Consider the behavior of | u k ( t ) | for all k = 1 , , ... . Define k u t k ≡ r ( t ) forsome r ( t ) >
0. Let us list all of possible cases: 1) b g ( t ) r ( t ) p ( t ) < b aλ + b b ( t ),2) ∃ λ k : b aλ k − + b b ( t ) < b g ( t ) r ( t ) p ( t ) < b aλ k + b b ( t )and 3) ∃ λ k : b g ( t ) r ( t ) p ( t ) = b aλ k + b b ( t ). Case 1) was already investigated,therefore we will consider here only cases 2) and 3).Consider either the case 2) or 3), i.e. ∃ λ k : b aλ k − + b b ( t ) < b g ( t ) r ( t ) p ( t ) < b aλ k + b b ( t )and ∃ λ k : b g ( t ) r ( t ) p ( t ) = b aλ k + b b ( t ) . We have the following system of equations0 = 12 ddt | u k ( t ) | + (cid:16)b aλ k + b b ( t ) (cid:17) | u k ( t ) | − b g ( t ) r ( t ) p ( t ) | u k ( t ) | == 12 ddt | u k ( t ) | + (cid:16)b aλ k + b b ( t ) − b g ( t ) r ( t ) p ( t ) (cid:17) | u k ( t ) | , (3.24) | u k ( t ) | = | u t k | , k = 1 , , ..., where u ( t, x ) ≡ ∞ P k =1 u k ( t ) w k ( x ). It is easy to see that this system of equations isof interest only for the cases k ≥ k , k ≤ k − k = k separately. In case 2) if k ≥ k this part of the system has a solution that is unique as t ≤ t ( k, r ( t )) forsome t ( k, r ( t )) > t . Formally, we can determine the solution of each equation from (3.24) to be(3.25) | u k ( t ) | = exp − t Z t (cid:16)b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) (cid:17) ds | u t k | . Thus, if we consider the expression (3.24) for k ≤ k − | u k ( t ) | == exp − t Z t (cid:16)b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) (cid:17) ds | u t k | ≥≥ exp n (cid:16)b g ( t ) r ( t ) p ( t ) − b aλ k − b b ( t ) (cid:17) ( t − t ) o | u t k | , as b g ( t ) r ( t ) p ( t ) > b aλ k + b b ( t ) for 1 ≤ k ≤ k − t > t . Consequently,the sequence | u k ( t ) | increases for each k : 1 ≤ k ≤ k − r ( t ) as long as (cid:13)(cid:13) P k t u t (cid:13)(cid:13) is sufficiently greater than (cid:13)(cid:13) Q k t u t (cid:13)(cid:13) .For case 3) for some k = k one has b g ( t ) r ( t ) p ( t ) = b aλ k + b b ( t ) for which | u k ( t ) | = exp − t Z t (cid:16)b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) (cid:17) ds | u t k | by virtue of (3.24). Here for the function ρ ( t ) = b aλ k + b b ( t ) − b g ( t ) r ( t ) p ( t ) we have ρ ( t ) = 0, but in general its variation is not known. Thus, it is impossible to obtaina monotonicity result for a solution to this equation, since the behavior of ρ ( t ) isnot known. As we shall explain in the sequel, the behavior of ρ ( t ) depends on thegeometrical properties of the initial data u on spheres S H r (0), 0 < r ≤ r ( t ).From the previously mentioned relationships it is clear that in order to investigatethe behavior of the functional ρ ( t ) one should study both k P k u k and k Q k u k .Clearly, if the condition 2) is assumed, then k P k u k increases, and k Q k u k decreasesas t > u = P k u + Q k u, we also find that k u k = k P k u k + k Q k u k . Thus, the behavior of the functional k u k ( t ) depends on the relationship betweenthe values k P k u t k and k Q k u t k . Let k u t k ≡ r ( t ) ; b aλ k − + b b ( t ) < b g ( t ) r ( t ) p ( t ) < b aλ k + b b ( t )and consider above equality, i.e.(3.26) k u k ( t ) = k P k u k ( t ) + k Q k u k ( t ) = X k ≤ k | u k ( t ) | + X k>k | u k ( t ) | . It is necessary to investigate the following 3 cases: a ) u t ≡ X k ≤ k − u t k w k ∈ P k (cid:0) H (Ω) (cid:1) ≡ H k ; b ) u t ≡ X k ≥ k u t k w k ∈ Q k (cid:0) H (Ω) (cid:1) ≡ H − k and c ) u t ≡ P k ≥ u t k w k , when c ) k Q k u t k < k P k u t k and c ) k Q k u t k ≥k P k u t k , separately.Consider a ): in this case we have | u k ( t ) | = exp − t Z t (cid:16)b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) (cid:17) ds | u t k | for any k = 1 , ..., k −
1, thus , u k ( t ) = 0 for k ≥ k since k u t k = r ( t ) ≡ P k ≤ k − ( u t k ) and r ( t ) ≡ P k ≤ k − | u k ( t ) | . On the other hand, in this case b aλ k + b b ( t ) − b g ( t ) r ( t ) p ( t ) < b g ( t ) r ( t ) p ( t ) > b aλ k + b b ( t ) for each k = 1 , ..., k − r ( t ) p ( t ) increases as t ↑ ∞ . EHAVIOR OF SOLUTIONS AND CHAOS 17
Now let us consider case b), i.e. k u t k = r ( t ) ≡ P k ≥ k ( u t k ) . Then | u k ( t ) | = exp − t Z t (cid:16)b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) (cid:17) ds | u t k | for any k ≥ k , giving rise to u k ( t ) = 0 for k = 1 , , ..., k − , since (cid:16)b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) (cid:17) > b g ( t ) r ( t ) p ( t ) < b aλ k + b b ( t ) for each k ≥ k and r ( t ) p ( t ) decreases as t ↑ ∞ . It follows from continuity that k u k ( t ) = r ( t )
Theorem 4.
Under the above conditions if (cid:13)(cid:13) u − t k (cid:13)(cid:13) H − k > (cid:13)(cid:13) u + t k (cid:13)(cid:13) H k and if therate of decrease of the norm (cid:13)(cid:13) u − k ( t ) (cid:13)(cid:13) H − k is greater than the rate of decrease ofthe norm (cid:13)(cid:13) u + k ( t ) (cid:13)(cid:13) H k , then there exists e t > t such that | u k ( t ) | decreases for t ≥ e t . On the other hand, if (cid:13)(cid:13) u − t k (cid:13)(cid:13) H − k < (cid:13)(cid:13) u + t k (cid:13)(cid:13) H k and the rate of increaserate of the norm (cid:13)(cid:13) u + k ( t ) (cid:13)(cid:13) H k is greater than the rate of decrease of (cid:13)(cid:13) u − k ( t ) (cid:13)(cid:13) H − k , then there exists b t > t such that | u k ( t ) | increases for t ≥ b t. Remark 6.
This shows how is need to control in order to change of the behaviorof the dynamics of cancer, i.e. to which coefficients or exponents is necessary tocontrol.
Now we will proceed to investigate the behavior of solutions beginning with thetime t when p ( t ) = 0 (i.e. remains only degregated cells) in detail. As were notedif one select a initial function u ∈ H then according of expression (3.10) on thetime t we have the function k u t k = r ( t ) == . exp − a λ t + t Z b ( s ) ds + t Z t (cid:16) b ( s ) − g ( s ) ( r ) p ( s ) (cid:17) ds k u k
28 KAMAL N. SOLTANOV
So, it isn’t difficult to see that arbitrarily chosen initial function u in the moment t will be u t that with the norm determined by the number r ( t ) will satisfies theinequalities b aλ k + b b ( t ) ≥ b g ( t ) r ( t ) p ( t ) > b aλ k − + b b ( t )for some λ k . As H = H k ⊕ H − k and for each u ∈ H one has the decomposition u ( t ) = u + k ( t ) + u − k ( t ) for any t >
0; then it suffices to study the case when (cid:13)(cid:13) u − t k (cid:13)(cid:13) H − k ≫ (cid:13)(cid:13) u + t k (cid:13)(cid:13) H k . Indeed if we set k u t k = r ( t ) and r ( t ) satisfies theinequality b aλ k + b b ( t ) > b g ( t ) r ( t ) p ( t ) > b aλ k − + b b ( t ) , then each of solutions to the problem (3.12) - (3.13) satisfies one of the followingstatements:1. If u t lies in H k or in a small neighborhood of the subspace H k , then | u k ( t ) | ↑ ∞ as t ↑ ∞ for k = 1 , k −
1, moreover since in this case r ( t ) increasesgradually and so in time b g ( t ) r ( t ) p ( t ) is greater than each b aλ k + b b ( t ) for k ≥ k ;i.e. in this case | u k ( t ) | gradually increases for all k ;2. If u t lies in H − k or in a small neighborhood of the subspace H − k , then | u k ( t ) | ↓ t ↑ ∞ for k ≥ k , moreover since in this case r ( t ) decreases graduallyso that in time b g ( t ) r ( t ) p ( t ) is less than each b aλ k + b b ( t ); i.e. in this case | u k ( t ) | gradually decreases for all k ;3. If u t ∈ H such that (cid:13)(cid:13) u − t k (cid:13)(cid:13) H − k ≈ (cid:13)(cid:13) u + t k (cid:13)(cid:13) H k , then there is a relationbetween of u − t k ( t ) and u + t k such that the behavior of the u k ( t ) is chaotic for all k for which u k (0) = 0.4. If u t ∈ H such that (cid:13)(cid:13) u − t k (cid:13)(cid:13) H − k and (cid:13)(cid:13) u + t k (cid:13)(cid:13) H k are different, then thesolutions are still connected by some relations.As the Claims 1 and 2 were proved above, we need only study Claim 3 and Claim4 Consider the representation of the formal solutions (3.25) to the problem (3.24): | u k ( t ) | = exp − t Z t (cid:16)b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) (cid:17) ds | u t k | ,k = 1 , , ... . It is known that | u k ( t ) | increases for k : 1 ≤ k ≤ k − k ≥ k by virtue of Theorem 4, depending on the difference b aλ k + b b ( t ) − b g ( t ) r ( t ) p ( t ) . But as r k ( t ) ≡ (cid:13)(cid:13) u − k (cid:13)(cid:13) H k ( t ) + (cid:13)(cid:13) u + k (cid:13)(cid:13) H − k ( t ) holds near t = t , r ( t ) can change depending on the behavior of | u k ( t ) | as k ≥ k and k ≤ k − H change, i.e. b g ( t ) r ( t ) p ( t ) canbecome greater than b aλ k + b b ( t ) or less than b aλ k − + b b ( t ). This variation of r ( t ) isvery complicated, as it depends on relations among the behaviors of u k ( t ) in thecase when k ≥ k and k ≤ k −
1, which may give rise to chaos.We now investigate Claim 4 in order to explain the question there is or not anattractor for the operator resolving the problem (3.12) - (3.13). Toward this end,we consider the following system of differential equations ddt h u ( t ) , w k i + h b a ∇ u ( t ) , ∇ w k i + b b ( t ) h u ( t ) , w k i − b g ( t ) k u ( t ) k p ( t ) h u ( t ) , w k i = 0 , where { w k ( x ) } ∞ k =1 are eigenfunctions of the Laplacian − ∆ in H (Ω) correspondingto the eigenvalues { λ k } ∞ k =1 , respectively, by virtue of the imposed conditions. EHAVIOR OF SOLUTIONS AND CHAOS 19
Whence, it follows that ddt u k ( t ) + (cid:16)b aλ k + b b ( t ) (cid:17) u k ( t ) − b g ( t ) k u ( t ) k p ( t ) u k ( t ) = 0 , k = 1 , , ... So, for the investigation of the behavior of solutions of the problem (3.12) - (3.13)under the condition u t ∈ S H (Ω) r (0), b aλ k − + b b ( t ) < b g ( t ) r p ( t )0 < b aλ k + b b ( t ) it isenough to study the following Cauchy problem:(3.27) ddt u k ( t ) + (cid:16)b aλ k + b b ( t ) (cid:17) u k ( t ) − b g ( t ) k u ( t ) k p ( t ) u k ( t ) = 0 , (3.28) h u ( t ) , w k i | t = t = u k ( t ) | t = t = u t k , k = 1 , , ..., k − t > t for t ∈ [ t , t ) we have b g ( t ) k u ( t ) k p ( t ) ≤ r p ( t )0 + ε < b aλ k + b b ( t ) , for some ε >
0. Indeed, we have from (3.14) - (3.15) that ddt u k ( t ) + h ( b aλ k + b b ( t ) − b g ( t ) k u ( t ) k p ( t ) i u k ( t ) = 0 ,u k ( t ) = u t k , which leads to the formal solution of the Cauchy problem(3.29) u k ( t ) = exp − t Z t ( b aλ k + b b ( s ) − b g ( s ) r ( s ) p ( s ) ) ds u t k Hence, if b aλ k − + b b ( t ) < b g ( t ) r p ( t )0 , then u k ( t, x ) increases in the vicinity of u t k ( x )if u t k ( x ) > k = 1 , , ..., k − k P k u t k is sufficiently greaterthan k Q k u t k .Let u t ∈ H (Ω) and k u t k = r , then the above expression implies that thebehavior of the solution u k ( t ) depends on the relationship between b g ( t ) r p ( t )0 and b aλ k + b b ( t ) and also between k P k u t k and k Q k u t k .Consider the behavior of | u k ( t ) | for all k = 1 , , ... , in the case when ∃ λ k : b aλ k − + b b ( t ) < b g ( t ) r p ( t )0 < b aλ k + b b ( t ). We have the following system ofequations ddt u k ( t ) + (cid:16)b aλ k − + b b ( t ) − b g ( t ) r p ( t )0 (cid:17) u k ( t ) , (3.30) u k ( t ) = 0 , u t k , k = 1 , , ..., where u ( t, x ) := ∞ P k =1 u k ( t ) w k ( x ). It is easy to see that this system of equations isof interest only for the cases k ≥ k , k ≤ k − k = k separately. If k ≥ k ,this part of the system has a solution that is unique as t ≤ t ( k, r ) for some t ( k, r ) > t if k Q k u t k is sufficiently greater than k P k u t k . Formally we candetermine the solution of each equation from (3.22)-(3.23) in the following form:(3.31) u k ( t ) = exp − t Z t b aλ k + b b ( s ) − b g ( s ) r p ( s )0 ds u t k . Thus, considering the problem (3.22)-(3.23) for k ≤ k −
1, (3.25) implies that | u k ( t ) | = exp − t Z t b aλ k + b b ( s ) − b g ( s ) r p ( s )0 ds | u t k | ≥≥ exp n(cid:16)b g ( t ) r p ( t )0 − b aλ k + b b ( t ) (cid:17) t o | u t k | , as b aλ k + b b ( t ) < b g ( t ) r p ( t )0 for 1 ≤ k ≤ k − t > t . Consequently,the sequence | u k ( t ) | increases for each k : 1 ≤ k ≤ k −
1, if the part k P k u t k issufficiently greater than k Q k u t k , and that renders the increase of r ( t ).Consider the representation of the formal solutions (3.25) to the problem (3.22)-(3.23): u k ( t ) = exp − t Z t b aλ k + b b ( s ) − b g ( s ) r p ( s )0 ds u t k , k = 1 , , .... It is known that | u k ( t ) | increases for k : 1 ≤ k ≤ k − k ≥ k by virtue of Theorem 4, depending on the difference b aλ k + b b ( t ) − b g ( t ) r p ( t )0 andthe relation between k P k u k and k Q k u k . Thus, concerning the system (3.25)for k we need to study the expression b aλ k + b b ( t ) − b g ( t ) r p ( t )0 which is negativedue to the conditions imposed. But the behavior of functions | u k ( t ) | cannot exactlyexplain the behavior of functions u k ( t ), and also the behavior of the solution u ( t, x ).Consequently, we need to study the behavior of functions u k ( t ) in greater detail.So, from the expression (3.25) under the corresponding relation between k P k u t k and k Q k u t k is clear that if u t k ≥ u k ( t ) ≥ b aλ k + b b ( t ) − b g ( t ) r p ( t )0 > u k ≥ u k ( t ) will decreasesat least in some vicinity of zero. And next let b aλ k + b b ( t ) − b g ( t ) r p ( t )0 < u t k ≥ u k ( t ) increases at least in a vicinity of zero. If b aλ k + b b ( t ) − b g ( t ) r p ( t )0 = 0 then u k ( t ) does not vary at least in some vicinity ofzero.Thus, we obviously need to investigate the behavior of r ( t ) for various function u t ( x ) in the case when k u t k = r , i.e. for various function u ( x ). Therefore, thebehavior of r ( t ) essentially depends on the selections of u k ( t ) and consequently,on the selections of initial function u .It is clear from the above analysis that we need to consider the expression u ( t, x ) = P k ≥ u k ( t ) w k ( x ) for the solution and the expression u t ( x ) = P k ≥ u t k w k ( x )for initial data at the moment t obtained from initial data u of examined problem(3.12)-(3.13). Let r >
0, then there exists k ≥ b aλ k − + b b ( t ) < b g ( t ) r p ( t )0 ≤ b aλ k + b b ( t )and k u t k = r .Using the orthogonal splitting u ( t ) = P k u ( t )+ Q k u ( t ), we obtain the followingexpressions: u t ( x ) = X k ≥ u t k w k ( x ) = X k − ≥ k ≥ u t k w k ( x ) + X k ≥ k u t k w k ( x ) EHAVIOR OF SOLUTIONS AND CHAOS 21 and(3.32) u ( t, x ) = X k ≥ u k ( t ) w k ( x ) = X k − ≥ k ≥ u k ( t ) w k ( x ) + X k ≥ k u k ( t ) w k ( x ) . There exist t > t and t > t such that P k u ( t ) of (3.32) can increases in( t , t ) and Q k u ( t ) of (3.32) decreases in ( t , t ). Let min { t , t } = t , then when t > t these summands can behave quite differently. Here are the possibilities: 1)the velocity k u k ( t ) becomes greater than r for t ≥ t , moreover b g ( t ) r ( t ) p ( t ) > b aλ k + b b ( t ) for t > t , so the orthogonal splitting u = P k u + Q k u changes andbecomes, at least, u = P k +1 u + Q k − u ; 2) Q k u decreases up to a point where k u k ( t ) is smaller than r for t ≥ t , moreover b g ( t ) r ( t ) p ( t ) < b aλ k − + b b ( t ) for t > t , so the orthogonal splitting u = P k u + Q k u changes and becomes, at least, u = P k − u + Q k +1 u ; 3) there exist a t ≥ t and an R ≥ k P k u k ≥ R > t the changes of P k u and Q k u become such that(3.33) r ( t ) = k u ( t ) k = X k − ≥ k ≥ | u k ( t ) | + X k ≥ k | u k ( t ) | satisfies R ≤ r ( t ) ≤ R for t ≥ t .Consider case 1) where we have the following possibilities: a) P k u increaseswith such velocity that k u k ( t ) ր ∞ , which can occur when u t ( x ) is chosen nearthe subspace H k (this scenario is studied in Proposition 3); b) the rate of growthof P k u diminishes beginning at time t and the function u ( t, x ) behaves as in case3, which we will explain in what follows. Case 2) has 2 variants: a’) Q k u decreaseswith such velocity that k u k ( t ) ց b g ( t ) r ( t ) p ( t ) < b aλ + b b ( t ), which can take place when u t ( x ) is chosen in near the subspace H − k (thisvariant is also studied in Proposition 3); b’) the rate of decrease of Q k u diminishesbeginning at some time t and leads to case 1)b).Consequently, it remains only to investigate case 3). It is clear that this occurs as P k u t positive. Therefore, we consider special initial data and try to explain case 3)for such functions. So, let P k u t = u t k − w k − , i.e. u t ( x ) = u t k − w k − ( x ) + Q k u t ( x ) and k u t k = r . Then we obtain the following: u k − ( t ) changes so that | u k − ( t ) | increases with t and b g ( t ) | u k − ( t ) | p ( t ) −→ b aλ k − + b b ( t ) when t ր ∞ ;moreover, k Q k u ( t ) k decreases with increasing t and therefore k Q k u ( t ) k −→ t ր ∞ . Hence, b g ( t ) k u ( t ) k p ( t ) ց b aλ k − + b b ( t ) as t ր ∞ . In other words,the increase of k P k u k and decrease of k Q k u ( t ) k compensate for each other in sucha way that this process leads to the case described above.Thus, it not is difficult to see that in order to obtain the above result, we needto select u t k − in near H − k , which depends on the given r : b aλ k − + b b ( t ) < b g ( t ) r p ( t )0 ≤ b aλ k + b b ( t ) . Accordingly it follows in the case when P k u t increases, the corresponding u t k ,1 ≤ k ≤ k −
1, must be chosen as done previously. Moreover, there is a λ j suchthat b g ( t ) k P k u ( t ) k p ( t ) ր b aλ j + b b ( t ) when t ր ∞ , where λ j = inf { λ k | ≤ k ≤ k − , u k = 0 } . Therefore, there exists a “double cone” with the “vertex at zero” that containsthe subspace H − k and all elements are contained in some neighborhood of H − k . In addition, the maximal distance between of the elements of this subset and the sub-space H − k depends on the given r . Now, we denote this subset by e H ⊂ H . It fol-lows from this definition that any subset of e H ∩ n B H r (0) − B H r (0) , r > r > o converges to a set, which we can define as H k ∩ B H λ j (0), where r , r are somenumbers with b aλ k − + b b ( t ) < b g ( t ) r p ( t )2 ≤ b aλ k + b b ( t )and there is a λ j = inf { λ k | ≤ k ≤ k − , u t k = 0 } and k = k ( r ). Thisshows that tthis isa subset of a finite-dimensional space and it is local attractor insome sense.Thus, we have proved the following result. Theorem 5.
Let all the imposed above conditions hold and u t ∈ H (Ω) whosenorm k u t k = r satisfies the inequality b aλ k − + b b ( t ) < b g ( t ) r p ( t )0 ≤ b aλ k + b b ( t ) . Then each solution to the problem (3.12) - (3.13) satisfies one of the followingproperties:1. If u t lies in H k or in a sufficiently small neighborhood of the subspace H k , then | u k ( t ) | ↑ ∞ as t ↑ ∞ for k = 1 , k − ; moreover, in this case r ( t ) = k u ( t ) k increases and b g ( t ) r ( t ) p ( t ) gradually becomes greater than b aλ k + b b ( t ) for each λ k : k ≥ k , | u k ( t ) | gradually increases for all k ;2. If u t lies in H − k or in a small neighborhood of the subspace H − k , then | u k ( t ) | ↓ as t ↑ ∞ for k ≥ k ; moreover, since in this case r ( t ) decreases and b g ( t ) r ( t ) p ( t ) gradually becomes less than b aλ k + b b ( t ) for each λ k , | u k ( t ) | graduallydecreases for all k ;3. If u t ∈ H , (cid:13)(cid:13) P k u t (cid:13)(cid:13) ≪ (cid:13)(cid:13) Q k u t (cid:13)(cid:13) and if there are small numbers δ (cid:16)b aλ k + b b ( t ) (cid:17) > ǫ (cid:16)b aλ k + b b ( t ) (cid:17) > such that the Hausdorff distance satisfies (3.34) ǫ ≤ d (cid:0) H − k ; (cid:8) u + t k (cid:12)(cid:12) k = 1 , k − (cid:9)(cid:1) ≤ δ, then u ( t, x ) is chaotic for sufficient large t . In addition, if (cid:13)(cid:13) u − t k (cid:13)(cid:13) H − k ≈ (cid:13)(cid:13) u + t k (cid:13)(cid:13) H k , then there is a relationship between u − t k ( t ) and u + t k for which the behavior of the u k ( t ) is chaotic for all k satisfying u k (0) = 0 . Remark 7.
If property 3 of the above theorem obtains, then the following claim isreasonable: for any λ k there is a subset B r ( λ k ) ⊂ H (Ω) for which (3.34) holdsand for any u t ∈ B r ( λ k ) the corresponding solution u ( t ) satisfies the condition b aλ j + b b ( t ) ≤ b g ( t ) r ( t ) p ( t ) < b aλ k + b b ( t ) λ k for any t > , and lim t ↑∞ b g ( t ) r ( t ) p ( t ) = b aλ j + b b ( t ) , u k = 0 then there is an absorbing chaotic set in L (Ω) , where λ j = inf { λ k | ≤ k ≤ k − } . The relation between u t and u is given by formula (3.10) EHAVIOR OF SOLUTIONS AND CHAOS 23
Remark 8.
These results shows that if is fulfilled the case p ( t ) = 0 then in thedomain Ω were unremained of the immune cells, moreover, since in this case thebehavior of the dynamics of the cancer is chaotic, therefore it become undefinable. References [1] Afraimovich, V., Babin, A. V., Chow, S. N. Spatial chaotic structure of attractors of reaction-diffusion systems. Trans. Am. Math. Soc. 348, (1996) 12, 5031–5063.[2] Agur Z., From the evolution of toxin resistance to virtual clinical trials: The role of mathe-matical models in oncology, Future Oncol., 6, (2010), 917-927.[3] Altrock Ph. M., Lin L. Liu, Michor F. The mathematics of cancer: integrating quantitativemodels, Nature Reviews: Cancer 15, (2015), 730-745.[4] Babin, A. V. Topological invariants and solutions with a high complexity for scalar semilinearPDE. J. Dynam. Differ. Equ. 12, (2000) 3, 599–646.[5] Barrachina X. and Conejero J. A. Devaney chaos and distributional chaos in the solution ofcertain partial differential equations. Abstract and Applied Analysis, (2012), 457019, 11 p.,doi:10.1155/2012/457019[6] Bekiros S., Kouloumpou D. SBDiEM: A new Mathematical model of Infectious DiseaseDynamics, Chaos, Solitons and Fractals (2020), doi:10.1016/j.chaos.2020.109828.[7] Bellomo, N., Preziosi, L. Modelling and mathematical problems related to tumor evolutionand its interaction with the immune system. Math. Comp. Modelling, 32, (2000), 413–452.[8] Bellomo, N., Li, N., Maini, P. On the foundations of cancer modeling: selected topics,speculations, and perspectives. Math. Mod. Methods Appl. Sci., 18(4), (2008), 593–646.[9] Berestycki, H., Larrouturou, B., Lions, P.L.: Multi-dimensional travelling wave solutions ofa flame propagation model. Arch. Rational Mech. Anal., 111, (1990), 33–49.[10] Blackmore D. The mathematical theory of chaos. Comp.&Maths. with Appls. 12B, (1986)N3/4, 1039-1045.[11] Britton, N.
Reaction-Diffusion Equations and Their Applications to Biology (1986), Aca-demic Press, Waltham, MA.[12] Bunimovich, L. A., Sinai, Ya. G. Space-time chaos in coupled map lattices. Nonlinearity 1,(1988) 4, 491–516.[13] Chaplain,M.
Modelling aspects of cancer growth: insight from mathematical and numer-ical analysis and computational simulation . In Lecture notes in mathematics: Vol. 1940,Multiscale problems in the life sciences, (2008), 147–200, Berlin: Springer.[14] Chaplain, M.A.J., Lolas, G.:Mathematical modelling of cancer invasion of tissue: Dynamicheterogeneity. Net. Hetero. Med. 1, (2006), 399–439.[ 15] Chueshov I.D. Theory of functionals that uniquely determine the asymptotic dynamicsof infinite-dimensional dissipative systems. Russian Mathematical Surveys, 53(4), (1998),731-776.[16] Chueshov I.D.
Introduction to the Theory of Infinite Dimensional Dissipative Systems .(1999) Kharkiv State University Publisher ACTA.[17] Collet, P., Eckmann, J.-P. The definition and measurement of the topological entropy perunit volume in parabolic PDEs, Nonlinearity 12, (1999) 3, 451–473.[ 18] Cordes H. O., Uber die erste randwertaufgabe bei quasilinearen differentialgleichungenzweiter ornung in mehr als zwei variablen, Math. Ann., 131, (1956), 278–312.[19] Crampin, E.J., Hackborn, W.W., Maini, P.K.: Pattern formation in reactiondiffusion modelswith nonuniform domain growth. Bull. Math. Biol., 64, (2002), 747–769.[20] David N., Benoit P. Free boundary limit of tumor growth model with nutrient,arXiv:2003.10731v2[math.AP]21Apr 2020[21] Dunford, N., Schwartz, J. T.
Linear Operators, General Theory . Pure and Applied Math.,(1957) VII, Intersci. Publ., Inc., New York, Jon Wiley&Sons, ix+858.[22] D¨uzg¨un F. G., Soltanov K. N., Existence, uniqueness and behaviour of solution for nonlineardiffusion type equation with third type boundary value, Rend. Lincei Mat. Appl., 26, (2015).[23] Eftimie R., Bramson J. L., Earn D. J. D. Interactions Between the Immune System andCancer: A Brief Review of Non-spatial Mathematical Models. Bull Math Biol 73, (2011),2–32.[24] Fiedler, B., Polacik, P. Complicated dynamics of scalar reaction diffusion equations with anonlocal term. Proc. Roy. Soc. Edinburgh, Sect. A, 115, (1990), 1–2, 167–192. [25] Friedman A.
Cancer Models and Their Mathematical Analysis , Lect. Notes Math. 1872,(2006), 223–246.[26] Friedman A., Bei Hu, Bifurcation from stability to instability for a free boundary problemmodeling tumor growth by Stokes equation. J. Math. Anal. Appl. 327, (2007), 643–664.[27] Grindrod, P.
The Theory and Applications of Reaction-Diffusion Equations , (1996), OxfordUniv. Press.[28] Henry, D.
Geometric theory of semilinear parabolic equations . Lecture Notes in Math. 840,(1981), Springer-Verlag, Berlin-Heidelberg-New York, vi+350.[29] Kalli K., Soltanov K. N., Existence and behavior of solutions for convection-diffusion equa-tions with third type boundary condition, TWMS J. Pure Appl. Math., 8, (2017), 2, 209-222[30] Kirchgassner, K. Nonlinear wave motion and homoclinic bifurcation. In
Theoretical andApplied Mechanics (Lyngby, 1984), North-Holland, Amsterdam, (1985), 219–231.[31] Lowengrub J. S., H B Frieboes, F Jin, Y-L Chuang, X Li, P Macklin, S M Wise, andV Cristini, Nonlinear modelling of cancer: bridging the gap between cells and tumours,Nonlinearity, 23(1), (2010), R1–R9.[32] Manneville, P. Dynamical systems, temporal vs. spatio-temporal chaos, and climate. In
Non-linear Dynamics and Pattern Formation in the Natural Environment (Noordwijkerhout,1994), 335, (1995) of Pitman Res. Notes Math. Ser., Longman, Harlow, 168–187.[33] Nagy, J. (2005) The ecology and evolutionary biology of cancer: a review of mathematicalmodels of necrosis and tumor cells diversity. Math. Biosci. Eng., 2(2), 381–418.[34] Nani F., Freedman H. I., A mathematical model of cancer treatment by immunotherapy,Math. Biosci., 163, (2000), 159-199.[35] Ozturk E., Soltanov, K. N., Solvability and longtime behavior of nonlinear reaction–diffusionequations with Robin boundary condition, Nonlinear Analysis: TMA, 108, (2014), 1-13.[36] Pesin, Y. B., Sinai, Ya. G. Space–time chaos in chains of weakly interacting hyperbolicmappings, In Adv. Soviet Math., (1991) Vol. 3,
Dynamical Systems and Statistical Mechanics (Moscow, 1991), Am. Math. Soc., Providence, RI, 165–198.[37] Polacik, P.
Parabolic equations: asymptotic behavior and dynamics on invariant manifolds .In Handbook of Dynamical Systems, (2002) Vol. 2, North-Holland, Amsterdam, 835–883.[38] Shangbin Cui, Friedman A., A Free Boundary Problem for a Singular System of DifferentialEquations: An Applicationto a Model of Tumor Growth. Transactions of the AmericanMathematical Society, 355, (2003), 9, 3537-3590.[39] Smoller, J.
Shock Waves and Reaction-Diffusion Equations , (1994), Springer-Verlag, NewYork.[40] Soltanov, K. N. On noncoercive semilinear equations. Nonlinear Analysis: Hybrid Systems2, (2008), 344–358.[41] Soltanov, K. N. Perturbation of the mapping and solvability theorems in the Banach space.Nonlinear Analysis: TMA, 72, (2010), 1, 164-175.[42] Soltanov, K. N. , Ahmedov M., Solvability of Equation of Prandtl-von Mises type, Theoremsof Embedding, Trans. NAS of Azerbaijan, Ser. Phys-Tech.&Math., 37, (2017), 1.[43] Soltanov, K. N. On some modification Navier–Stokes equations, Nonlinear Analysis TMA,52, (2003), 769 – 793.[44 ] Soltanov K. N., Prikarpatski A. K., Blackmore D., Long-time behavior of solutions and chaosin reaction-diffusion equations, Chaos,Solitons and Fractals, 99, (2017), 91-100.[45] Soltanov K. N., Sert U., Certain results for a class of nonlinear functional spaces, CarpathianMath. Publ. (2020), 12 (1), 208–228.[46] Stuart, J. On the nonlinear mechanism of wave disturbances in stable and unsable parallelflows. Part I.. J. Fluid Mech., 9, (1960), 353–370.[47] Volpert, V., Petrovskii, S. Reaction-diffusion waves in biology, Phys. of Life Rev. 6, (2009),267-310.[48] Zeidler, E.
Nonlinear Functional Analysis and Applications , (1990) II/A-B, Springer-VerlagNew York Inc. vii+1202[49] Zelik, S. V. Spatial and dynamical chaos generated by reaction diffusion systems in un-bounded domains. J. Dynam. Differ. Eqs. 19, (2007), 1, 1-74.[50] Zelik, S. V., Mielke, A.
Multi-pulse evolution and space–time chaos in dissipative systems .Mem. Amer. Math. Soc. (2009) 198, 925, vi+97.
EHAVIOR OF SOLUTIONS AND CHAOS 25
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