The Cauchy Problem for a One Dimensional Nonlinear Peridynamic Model
aa r X i v : . [ m a t h . A P ] J a n The Cauchy Problem for a One Dimensional Nonlinear ElasticPeridynamic Model
H. A. Erbay , A. Erkip , G. M. Muslu ∗ Faculty of Arts and Sciences, Ozyegin University, Cekmekoy 34794, Istanbul, Turkey Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey Department of Mathematics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey
Abstract
This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamicmodel describing the dynamic response of an infinitely long elastic bar. The issues oflocal well-posedness and smoothness of the solutions are discussed. The existence of aglobal solution is proved first in the sublinear case and then for nonlinearities of degreeat most three. The conditions for finite-time blow-up of solutions are established.
Keywords:
Nonlocal Cauchy problem, Nonlinear peridynamic equation, Globalexistence, Blow-up.
1. Introduction
In this study, we consider the one-dimensional nonlinear nonlocal partial differentialequation, arising in the peridynamic modelling of an elastic bar, u tt = Z R α ( y − x ) w ( u ( y, t ) − u ( x, t )) dy, x ∈ R , t > u ( x,
0) = ϕ ( x ) , u t ( x,
0) = ψ ( x ) . (1.2)In (1.1)-(1.2) the subscripts denote partial differentiation, u = u ( x, t ) is a real-valuedfunction, the kernel function α is an integrable function on R and w is a twice differen-tiable nonlinear function with w (0) = 0. We first establish the local well-posedness of theCauchy problem (1.1)-(1.2), considering four different cases of initial data: (i) continu-ous and bounded functions, (ii) bounded L p functions (1 ≤ p ≤ ∞ ), (iii) differentiableand bounded functions and (iv) L p functions whose distributional derivatives are also in ∗ Corresponding author. Tel: +90 212 285 3257 Fax: +90 212 285 6386
Email addresses: [email protected] (H. A. Erbay ), [email protected] (A.Erkip ), [email protected] (G. M. Muslu ) Preprint submitted to Journal of Differential Equations September 25, 2018 p ( R ). We then extend the results to the case of L Sobolev spaces of arbitrary (non-integer) order for the particular form w ( η ) = η . We prove global existence of solutionsfor two types of nonlinearities: when w ( η ) is sublinear and when w ( η ) = | η | ν − η for ν ≤
3. Lastly, for the general case we provide the conditions under which the solutionsof the Cauchy problem blow-up in finite time.Equation (1.1) is a model proposed to describe the dynamical response of an infinitehomogeneous elastic bar within the context of the peridynamic formulation of elasticitytheory. The peridynamic theory of solids, mainly proposed by Silling [1], is an alterna-tive formulation for elastic materials and has attracted attention of a growing numberof researchers. The most important feature of the peridynamic theory is that the forceacting on a material particle, due to interaction with other particles, is written as afunctional of the displacement field. This means the peridynamic theory is a nonlocalcontinuum theory and regarding nonlocality it bears a strong resemblance to more tradi-tional theories of nonlocal elasticity, which are principally based on integral constitutiverelations [2, 3, 4]. As in other nonlocal theories of elasticity, the main motivation isto propose a generalized elasticity theory that involves the effect of long-range internalforces of molecular dynamic, neglected in the conventional theory of elasticity. Anotherfeature of the peridynamic theory is that the peridynamic equation of motion does notinvolve spatial derivatives of the displacement field. The absence of spatial derivativesof the displacement field in the equation of motion makes possible to use the peridy-namic equations even at points of displacement discontinuity. Furthermore, in contrastto the conventional theory of elasticity, the peridynamic theory predicts dispersive wavepropagation as a property of the medium even if the geometry does not define a lengthscale.In the peridynamic theory, by assuming a uniform cross-section and the absence ofbody forces, the governing equation of an infinitely long elastic bar is given by ρ u tt = Z R f ( u ( y, t ) − u ( x, t ) , y − x ) dy, (1.3)where the axis of the bar coincides with the coordinate axis, a material point on the axis ofthe bar has coordinate x in the undeformed state, u and f may be interpreted as averagesof the axial displacement and the axial force located at any x at time t , taken over a crosssection of the bar, and ρ is density of the bar material [5, 6]. The space integral in (1.3)implies that the displacement at a generic point is influenced by the displacements of allparticles of the bar (As commonly known, in the conventional theory of elasticity, theequation governing the dynamic response of an infinitely long bar is a hyperbolic partialdifferential equation that does not involve such a space integral originating from thenonlocal character of the peridynamic theory). Equation (1.3) is obtained by integratingthe equation of motion for the axial displacement over the cross-section and dividingthrough by the area of the cross-section. The bar is supposed to be composed of ahomogeneous objective microelastic material [1, 5, 6] and its constitutive behavior isdescribed by the function f . Newton’s third law imposes the following restriction on theform of f : f ( η, ζ ) = − f ( − η, − ζ ) (1.4)for all relative displacements η = u ( y, t ) − u ( x, t ) and relative positions ζ = y − x . For a2inear peridynamic material the constitutive relation is given by f ( η, ζ ) = α ( ζ ) η where α is called the micromodulus function [1, 5, 6]. It follows from (1.4) that α mustbe an even function. In [5, 6] the dynamic response of a linear peridynamic bar has beeninvestigated and some striking observations that are not found in the classical theory ofelastic bars have been made. Some results on the well-posedness of the Cauchy problemfor the linear peridynamic model have been established in [7, 8, 9, 10]. In spite of itsage, there is quite extensive literature on the linear peridynamic theory.It is natural to think that more interesting behavior may be observed when theattention is confined to the nonlinear peridynamic materials. From this point of view, tothe best of our knowledge, the present study appears to be the first study on mathematicalanalysis of nonlinear peridynamic equations. Techniques similar to those in [11, 12,13] enable us to answer some basic questions, like local well-posedness and lifespan ofsolutions, as the groundwork of further analysis of the nonlinear peridynamic problem.In this study we consider the case in which the constitutive behavior is described bya class of nonlinear peridynamic models in the separable form: f ( η, ζ ) = α ( ζ ) w ( η ) (1.5)where α and w are two functions satisfying the restriction imposed by (1.4). This sepa-rable form, while allowing us to exploit the properties of convolution-based techniques, isnot a serious restriction and it just makes the proofs easier to follow. Our results can becarried over to the case of general f ( η, ζ ). We illustrate this in Theorem 2.8; by imposingcertain differentiability and integrability conditions on f , we prove local well-posednessfor the general nonlinear peridynamic problem. Throughout this study we assume that α is an integrable even function while w is a differentiable odd function so that (1.4) issatisfied.Substitution of the separable form of (1.5) into (1.3) and non-dimensionalization ofthe resulting equation (or simply taking the mass density to be 1) gives the governingequation of the problem in its final form (1.1) (Henceforth we use non-dimensional quan-tities but for convenience use the same symbols). The aim of this study is three-fold: toestablish the local well-posedness of the Cauchy problem, to investigate the existence ofa global solution, and to present the conditions for finite-time blow-up of solutions.The paper is organized as follows. In Section 2, the existence and uniqueness ofthe local solution for the nonlinear Cauchy problem is proved by using the contractionmapping principle. For initial data in fractional Sobolev spaces the general case seems toinvolve technical difficulties and in Section 3 we consider the particular case w ( η ) = η inthe L Sobolev space setting. We note that the cubic case can be easily generalized to anarbitrary polynomial of η . In Section 4, we consider the issue of global existence versusfinite time blow-up of solutions. We first show that blow-up must necessarily occur in the L ∞ -norm. We then prove two results on global existence and finally establish blow-upcriteria.Throughout this paper, C denotes a generic constant. We use k u k ∞ and k u k p to denote the norms in L ∞ ( R ) and L p ( R ) spaces, respectively. The notation h u, v i denotes the inner product in L ( R ). Furthermore, C b ( R ) denotes the space of continuousbounded functions on R , and C b ( R ) is the space of differentiable functions in C b ( R )3hose first-order derivatives also belong to C b ( R ). In the spaces C b ( R ) and C b ( R ) wehave the norms k u k ∞ and k u k ,b = k u k ∞ + k u ′ k ∞ , respectively, where the symbol ′ denotes the differentiation. The Sobolev space W ,p ( R ) is the space of L p functionswhose distributional derivatives are also in L p ( R ) with norm k u k W ,p = k u k p + k u ′ k p .Similarly, for integer k ≥ C kb ( R ) denotes the space of functions whose derivatives up toorder k are continuous and bounded; W k,p ( R ) denotes the space of L p functions whosederivatives up to order k are in L p ( R ).
2. Local Well Posedness
Below we will give several versions of local well-posedness of the nonlinear Cauchyproblem given by (1.1)-(1.2). This is achieved in Theorems 2.2-2.5 for four different casesof initial data spaces, namely C b ( R ), L p ( R ) ∩ L ∞ ( R ), C b ( R ) and W ,p ( R ) (1 ≤ p ≤ ∞ ).The proofs will follow the same scheme given below.If (1.1) is integrated twice with respect to t , the solution of the Cauchy problemsatisfies the integral equation u = Su where( Su )( x, t ) = ϕ ( x ) + tψ ( x ) + Z t ( t − τ )( Ku )( x, τ ) dτ, (2.1)with ( Ku )( x, t ) = Z R α ( y − x ) w ( u ( y, t ) − u ( x, t )) dy. (2.2)Let X be the Banach space with norm k . k X , where the initial data lie. We thendefine the Banach space X ( T ) = C ([0 , T ] , X ), endowed with the norm k u k X ( T ) =max t ∈ [0 ,T ] k u ( t ) k X , and the closed R -ball Y ( T ) = { u ∈ X ( T ) : k u k X ( T ) ≤ R } . Wewill show that for suitably chosen R and sufficiently small T , the map S is a contractionon Y ( T ). This will be achieved by estimating first Ku and then Su in appropriate norms.In each of the four cases, for u, v ∈ Y ( T ) we will get estimates of the form k Su k X ( T ) ≤ k ϕ k X + T J ( R, T ) (2.3) (cid:13)(cid:13)(cid:13)(cid:13)Z t ( t − τ )(( Ku )( τ ) − ( Kv )( τ )) dτ (cid:13)(cid:13)(cid:13)(cid:13) X ( T ) ≤ T J ( R, T ) k u − v k X ( T ) (2.4)and hence k Su − Sv k X ( T ) ≤ T J ( R, T ) k u − v k X ( T ) (2.5)with certain functions J and J nondecreasing in R and T . Taking R ≥ k ϕ k X andthen choosing T small enough to satisfy T J ( R, T ) ≤ R/ S : Y ( T ) → Y ( T );the further choice T J ( R, T ) ≤ / S is a contraction. This implies thatthere is a unique u ∈ Y ( T ) satisfying the integral equation u = Su . But, as Ku is clearlycontinuous in t , we can differentiate (2.1) to get u t ( x, t ) = ψ ( x ) + Z t ( Ku )( x, τ ) dτ u tt ( x, t ) = ( Ku )( x, t ). This shows that u ∈ C ([0 , T ] , X ) solves (1.1)-(1.2). Finally, if u and u satisfy (1.1)-(1.2) with initial data ϕ i , ψ i for i = 1 , u − u = ϕ − ϕ + t ( ψ − ψ ) + Z t ( t − τ )(( Ku )( τ ) − ( Ku )( τ )) dτ. Then the estimate (2.4) shows that k u − u k X ( T ) ≤ k ϕ − ϕ k X + t k ψ − ψ k X + T J ( R, T ) k u − u k X ( T ) . When
T J ( R, T ) ≤ / k u − u k X ( T ) ≤ k ϕ − ϕ k X + 2 t k ψ − ψ k X for t ∈ [0 , T ]. This shows that, locally, solutions of (1.1)-(1.2) depend continuously oninitial data; thus the problem (1.1)-(1.2) is locally well posed.The Mean Value Theorem for nonlinear estimates and the following lemma for con-volution estimates will be our main tools: Lemma 2.1.
Let ≤ p ≤ ∞ and f ∈ L ( R ) , g ∈ L p ( R ) . The convolution ( f ∗ g )( x ) = R R f ( y − x ) g ( y ) dy is well defined and f ∗ g ∈ L p ( R ) with k f ∗ g k p ≤ k f k k g k p . In the estimates below, we will often encounter the nondecreasing function M ( R )defined for R > M ( R ) = max | η |≤ R | w ′ ( η ) | . (2.6)We now state and prove (i.e. show that the estimates (2.3) and (2.5) hold) the fourtheorems of local well posedness. Theorem 2.2.
Assume that α ∈ L ( R ) and w ∈ C ( R ) with w (0) = 0 . Then there issome T > such that the Cauchy problem (1.1)-(1.2) is well posed with solution in C ([0 , T ] , C b ( R )) for initial data ϕ, ψ ∈ C b ( R ) .Proof. Take X = C b ( R ). For u ∈ Y ( T ), clearly Ku is continuous in x and t and hence Su ∈ C ([0 , T ] , X ). Since w (0) = 0 and | u ( y, t ) − u ( x, t ) | ≤ k u ( t ) k ∞ , the Mean Value Theorem implies | w ( u ( y ) − u ( x )) | ≤ sup | η |≤ k u k ∞ | w ′ ( η ) | | u ( y ) − u ( x ) | = M ( k u k ∞ )( | u ( y ) | + | u ( x ) | ) , where we have suppressed the t variable for convenience. Then | ( Ku )( x, t ) | ≤ M ( k u ( t ) k ∞ ) Z R | α ( y − x ) | ( | u ( y, t ) | + | u ( x, t ) | ) dy = M ( k u ( t ) k ∞ ) [( | α | ∗ | u | ) ( x, t ) + k α k | u ( x, t ) | ] , (2.7)5nd k ( Ku )( t ) k ∞ ≤ M ( k u ( t ) k ∞ ) k α k k u ( t ) k ∞ (2.8)where we have used Lemma 2.1. Then | ( Su )( x, t ) | ≤ | ϕ ( x ) | + t | ψ ( x ) | + Z t ( t − τ ) | ( Ku )( x, τ ) | dτ, and k ( Su )( t ) k ∞ ≤ k ϕ k ∞ + t k ψ k ∞ + 2 k α k Z t ( t − τ ) M ( k u ( τ ) k ∞ ) k u ( τ ) k ∞ dτ. (2.9)As u ∈ Y ( T ), this gives M ( k u ( τ ) k ∞ ) ≤ M ( R ) and hence k Su k X ( T ) ≤ k ϕ k ∞ + T k ψ k ∞ + 2 M ( R ) k α k k u k X ( T ) sup t ∈ [0 ,T ] Z t ( t − τ ) dτ ≤ k ϕ k ∞ + T k ψ k ∞ + M ( R ) R k α k T . (2.10)This proves (2.3) with J ( R, T ) = k ψ k ∞ + M ( R ) R k α k T . Now let u, v ∈ Y ( T ). We startby estimating Ku − Kv . Again suppressing t , | w ( u ( y ) − u ( x )) − w ( v ( y ) − v ( x )) | ≤ M ( R )( | u ( y ) − v ( y ) | + | u ( x ) − v ( x ) | ) , and | ( Ku )( x, t ) − ( Kv )( x, t ) | ≤ M ( R ) ( | α | ∗ | u − v | ) ( x, t )+ M ( R ) k α k | u ( x, t ) − v ( x, t ) | . (2.11)Similar to (2.10) we get k ( Su )( t ) − ( Sv )( t ) k ∞ ≤ M ( R ) k α k Z t ( t − τ ) k u ( τ ) − v ( τ ) k ∞ dτ (2.12)and k Su − Sv k X ( T ) ≤ M ( R ) k α k T k u − v k X ( T ) (2.13)which proves (2.5) with J ( R, T ) = M ( R ) k α k T . According to the scheme describedabove, this completes the proof. Theorem 2.3.
Let ≤ p ≤ ∞ . Assume that α ∈ L ( R ) and w ∈ C ( R ) with w (0) = 0 .Then there is some T > such that the Cauchy problem (1.1)-(1.2) is well posed withsolution in C ([0 , T ] , L p ( R ) ∩ L ∞ ( R )) for initial data ϕ, ψ ∈ L p ( R ) ∩ L ∞ ( R ) .Proof. Let X = L p ( R ) ∩ L ∞ ( R ) with norm k u k X = k u k p + k u k ∞ . As we already have the L ∞ estimates given in (2.9) and (2.12), we now look for the corresponding L p estimates.Lemma 2.1 implies k ( | α | ∗ | u | ) ( t ) k p ≤ k α k | k u ( t ) k p so k ( Ku )( t ) k p ≤ M ( k u ( t ) k ∞ ) k α k k u ( t ) k p (2.14)6nd Minkowski’s inequality for integrals will yield k ( Su )( t ) k p ≤ k ϕ k p + t k ψ k p + 2 k α k Z t ( t − τ ) M ( k u ( τ ) k ∞ ) k u ( τ ) k p dτ. (2.15)Adding this to the L ∞ estimate (2.9), we get k Su k X ( T ) ≤ k ϕ k X + T k ψ k X + M ( R ) R k α k T . Similarly we have k ( Su )( t ) − ( Sv )( t ) k p ≤ M ( R ) k α k Z t ( t − τ ) k u ( τ ) − v ( τ ) k p dτ. (2.16)Adding this to (2.12) gives k Su − Sv k X ( T ) ≤ M ( R ) k α k T k u − v k X ( T ) and concludes the proofs of (2.3) and (2.5). Theorem 2.4.
Assume that α ∈ L ( R ) and w ∈ C ( R ) with w (0) = 0 . Then there issome T > such that the Cauchy problem (1.1)-(1.2) is well posed with solution in C ([0 , T ] , C b ( R )) for initial data ϕ, ψ ∈ C b ( R ) .Proof. We now take X = C b ( R ) for which the norm is k u k ,b = k u k ∞ + k u ′ k ∞ . Sincewe have the sup norm estimates (2.9) and (2.12) all we need is estimates for their x derivatives. Throughout this proof we will suppress t (or τ ) to keep the expressionsshorter, whenever it is clear from the context. Differentiating (2.2) gives ∂∂x ( Ku )( x ) = ∂∂x Z R α ( y − x ) w ( u ( y ) − u ( x )) dy = ∂∂x Z R α ( z ) w ( u ( x + z ) − u ( x )) dz = Z R α ( z ) w ′ ( u ( x + z ) − u ( x ))( u x ( x + z ) − u x ( x )) dz = Z R α ( y − x ) w ′ ( u ( y ) − u ( x ))( u x ( y ) − u x ( x )) dy. Recall that | w ′ ( u ( y ) − u ( x )) | ≤ M ( k u ( t ) k ∞ ) due to (2.6). Then | ( Ku ) x ( x ) | ≤ M ( k u k ∞ ) Z R | α ( y − x ) | ( | u x ( y ) | + | u x ( x ) | ) dy ≤ M ( k u k ∞ )[( | α | ∗ | u x | ) ( x ) + k α k | u x ( x ) | ] . (2.17)Since | ( Su ) x ( x, t ) | ≤ | ϕ ′ ( x ) | + t | ψ ′ ( x ) | + Z t ( t − τ ) | ( Ku ) x ( x, τ ) | dτ, (2.18)we have k ( Su ) x ( t ) k ∞ ≤ k ϕ ′ k ∞ + t k ψ ′ k ∞ + 2 k α k Z t ( t − τ ) M ( k u ( τ ) k ∞ ) k u x ( τ ) k ∞ dτ. M ( k u ( τ ) k ∞ ) ≤ M ( R ) so adding up with the estimate (2.9) proves (2.3) k Su k X ( T ) = max t ∈ [0 ,T ] ( k ( Su )( t ) k ∞ + k ( Su ) x ( t ) k ∞ ) ≤ k ϕ k ,b + T k ψ k ,b + M ( R ) R k α k T . Next, for | η i | ≤ R and | µ i | ≤ R for ( i = 1 , | w ′ ( η ) µ − w ′ ( η ) µ | ≤ | w ′ ( η ) | | µ − µ | + | w ′ ( η ) − w ′ ( η ) | | µ |≤ M ( R ) | µ − µ | + 2 R max η ≤ R | w ′′ ( η ) | | η − η |≤ M ( R ) | µ − µ | + 2 RN ( R ) | η − η | where N ( R ) = max η ≤ R | w ′′ ( η ) | . Then | ( Ku − Kv ) x ( x ) | ≤ M ( R ) Z R | α ( y − x ) | ( | u x ( y ) − v x ( y ) | + | u x ( x ) − v x ( x ) | ) dy +2 RN ( R ) Z R | α ( y − x ) | ( | u ( y ) − v ( y ) | + | u ( x ) − v ( x ) | ) dy ≤ M ( R )(( | α | ∗ | u x − v x | ) ( x ) + k α k | u x ( x ) − v x ( x ) | )+2 RN ( R )(( | α | ∗ | u − v | ) ( x ) + k α k | u ( x ) − v ( x ) | ) (2.19)and k ( Su − Sv ) x ( t ) k ∞ ≤ M ( R ) k α k Z t ( t − τ ) k u x ( τ ) − v x ( τ ) k ∞ dτ +4 RN ( R ) k α k Z t ( t − τ ) k u ( τ ) − v ( τ ) k ∞ dτ ≤ ( M ( R ) + 2 RN ( R )) k α k T k u − v k X ( T ) (2.20)Finally, adding this to (2.12) we get (2.5) in the form k Su − Sv k X ( T ) ≤ max t ∈ [0 ,T ] ( k ( Su − Sv )( t ) k ∞ + k ( Su − Sv ) x ( t ) k ∞ ) ≤ M ( R ) + RN ( R )) k α k T k u − v k X ( T ) . Theorem 2.5.
Let ≤ p ≤ ∞ . Assume that α ∈ L ( R ) and w ∈ C ( R ) with w (0) = 0 .Then there is some T > such that the Cauchy problem (1.1)-(1.2) is well posed withsolution in C ([0 , T ] , W ,p ( R )) for initial data ϕ, ψ ∈ W ,p ( R ) .Proof. Let X = W ,p ( R ) ⊂ L ∞ ( R ). Since k u k W ,p = k u k p + k u ′ k p , we need derivativeestimates only in addition to the L p estimates (2.15) and (2.16). For u, v ∈ Y ( T ), from(2.17)-(2.18) and Minkowski’s inequality we have k ( Su ) x ( t ) k p ≤ k ϕ ′ k p + t k ψ ′ k p + 2 k α k Z t ( t − τ ) M ( k u ( τ ) k ∞ ) k u x ( τ ) k p dτ.
8e note that the term k u k ∞ can be eliminated by using k u k ∞ ≤ C k u k W ,p due to theSobolev Embedding Theorem. So M ( k u ( τ ) k ∞ ) ≤ M ( CR ) and adding up the aboveestimate with (2.15) proves (2.3); k Su k X ( T ) = max t ∈ [0 ,T ] ( k ( Su )( t ) k p + k ( Su ) x ( t ) k p ) ≤ k ϕ k W ,p + T k ψ k W ,p + M ( CR ) R k α k T . Again from (2.19) we get k ( Su − Sv ) x ( t ) k p ≤ M ( CR ) k α k Z t ( t − τ ) k u x ( τ ) − v x ( τ ) k p dτ +4 RN ( CR ) k α k Z t ( t − τ ) k u ( τ ) − v ( τ ) k p dτ. Together with (2.16), we conclude the proof: k Su − Sv k X ( T ) ≤ M ( CR ) + RN ( CR )) k α k T k u − v k X ( T ) . Remark 2.6.
We remark that the investigation can also continue for smoother data inalong the same lines. That is, for initial data in C kb ( R ) or W k,p ( R ) with integer k we canprove higher-order versions of Theorems 2.4-2.5. Also, the proofs clearly indicate that inTheorems 2.2 and 2.3 we can replace the assumption w ∈ C ( R ) with its weaker form: w is locally Lipschitz. Similarly, in Theorems 2.4 and 2.5 the assumption w ∈ C ( R ) canbe weakened to the condition: w ′ is locally Lipschitz. Remark 2.7.
The above theorems of local well-posedness can be easily adapted to thegeneral peridynamic equation (1.3). Theorem 2.8 below extends Theorem 2.2 to the gen-eral peridynamic equation (1.3). Clearly, similar extensions are also possible in the casesof Theorems 2.3-2.5.
Theorem 2.8.
Assume that f ( ζ,
0) = 0 and f ( ζ, η ) is continuously differentiable in η for almost all ζ . Moreover, suppose that for each R > , there are integrable functions Λ R , Λ R satisfying | f ( ζ, η ) | ≤ Λ R ( ζ ) , | f η ( ζ, η ) | ≤ Λ R ( ζ ) for almost all ζ and for all | η | ≤ R . Then there is some T > such that the Cauchyproblem (1.3)-(1.2) is well posed with solution in C ([0 , T ] , C b ( R )) for initial data ϕ, ψ ∈ C b ( R ) .Proof. We proceed as in the proof of Theorem 2.2. By the Dominated ConvergenceTheorem, the condition | f ( ζ, η ) | ≤ Λ R ( ζ ) implies that Ku is continuous in x so that S : X ( T ) → X ( T ). Using the second inequality | f η ( ζ, η ) | ≤ Λ R ( ζ ) the estimates for k ( Su )( t ) k ∞ and k ( Su )( t ) − ( Sv )( t ) k ∞ follow as in (2.10) and (2.13), just replacing theterm M ( R ) k α k by k Λ R k and k Λ R k respectively, completing the proof.9 emark 2.9. To finish this section let us briefly mention the issue of multidimensionalcase in the general three-dimensional peridynamic theory. Although our analysis in thissection has been presented for the one-dimensional case of the peridynamic formulation,the techniques used can be extended to the case of a system of three peridynamic equationsin three space variables without any additional complication. Namely, if we replace thescalars x , y , u , w and α in (1.1)-(1.2) by the vectors x , y , u , w ( u ) and the matrix α ( x ) ,respectively, the local existence theorems given above will still be valid.
3. The Cubic Nonlinear Case in H s ( R ) We now want to consider the Cauchy problem (1.1)-(1.2) in the L Sobolev spacesetting. We will denote the L Sobolev space of order s on R by H s ( R ) with norm k u k H s = Z R (1 + ξ ) s | b u ( ξ ) | d ξ where b u denotes the Fourier transform of u . For integer k ≥ H k ( R ) = W k, ( R ).As mentioned in Remark 2.6, the proof in the case of H ( R ) can be extended to H k ( R ). On the other hand, for non-integer s , H s estimates of the nonlinear term w ( u ( y ) − u ( x )) involve technical difficulties. Nevertheless, the case of polynomial nonlinearities canbe handled in a straightforward manner. We illustrate this in the typical case w ( η ) = η .Then, the integral on the right-hand side of (1.1) can be computed explicitly in terms ofconvolutions and the Cauchy problem (1.1)-(1.2) becomes u tt = α ∗ u − u ( α ∗ u ) + 3 u ( α ∗ u ) − Au (3.1) u ( x,
0) = ϕ ( x ) , u t ( x,
0) = ψ ( x ) , (3.2)where A = R R α ( y ) dy .For the estimates below we need the following lemmas. Lemma 3.1.
Let α ∈ L ( R ) and u ∈ H s ( R ) for s ≥ . Then α ∗ u ∈ H s ( R ) and k α ∗ u k H s ≤ k α k k u k H s . Lemma 3.2. [14] Let s ≥ and u, v ∈ H s ( R ) ∩ L ∞ ( R ) . Then uv ∈ H s ( R ) and for someconstant C (independent of u and v ) k uv k H s ≤ C ( k u k ∞ k v k H s + k v k ∞ k u k H s ) . For the space H s ( R ) ∩ L ∞ ( R ) we use the norm k u k s, ∞ = k u k H s + k u k ∞ . In general,Lemma 3.2 implies that H s ( R ) ∩ L ∞ ( R ) is an algebra k uv k s, ∞ ≤ C k u k s, ∞ k v k s, ∞ , (3.3)and, by Lemmas 2.1 and 3.1, for α ∈ L ( R ) k α ∗ u k s, ∞ ≤ k α k k u k s, ∞ . (3.4)We are now ready to prove the following theorem.10 heorem 3.3. Let s > . Assume that ϕ, ψ ∈ H s ( R ) ∩ L ∞ ( R ) . Then there issome T > such that the Cauchy problem (3.1)-(3.2) is well posed with solution in C ([0 , T ] , H s ( R ) ∩ L ∞ ( R )) .Proof. We follow the scheme summarized at the beginning of Section 2 for X = H s ( R ) ∩ L ∞ ( R ). Explicitly, Ku = α ∗ u − u ( α ∗ u ) + 3 u ( α ∗ u ) − Au . We start by estimating the terms of the form u i ( α ∗ u j ) for i + j = 3. Clearly from (3.3)and (3.4), (cid:13)(cid:13) u i (cid:0) α ∗ u j (cid:1)(cid:13)(cid:13) s, ∞ ≤ C k α k k u k s, ∞ . Nevertheless, for later use we derive a moreprecise estimate. By repeated use of Lemma 3.2 we have (cid:13)(cid:13) u j (cid:13)(cid:13) H s ≤ C j k u k j − ∞ k u k H s .Again, by Lemmas 3.1 and 3.2 (cid:13)(cid:13) u i (cid:0) α ∗ u j (cid:1)(cid:13)(cid:13) H s ≤ C ( (cid:13)(cid:13) u i (cid:13)(cid:13) H s (cid:13)(cid:13) α ∗ u j (cid:13)(cid:13) ∞ + (cid:13)(cid:13) u i (cid:13)(cid:13) ∞ (cid:13)(cid:13) α ∗ u j (cid:13)(cid:13) H s ) ≤ C ( C i + C j ) k α k k u k ∞ k u k H s , so that k Ku k s, ∞ ≤ C k α k k u k ∞ k u k s, ∞ . Similarly (cid:13)(cid:13) u i ( α ∗ u j ) − v i ( α ∗ v j ) (cid:13)(cid:13) s, ∞ ≤ (cid:13)(cid:13) u i ( α ∗ ( u j − v j )) (cid:13)(cid:13) s, ∞ + (cid:13)(cid:13) ( u i − v i )( α ∗ v j ) (cid:13)(cid:13) s, ∞ ≤ C (cid:16)(cid:13)(cid:13) u i (cid:13)(cid:13) s, ∞ (cid:13)(cid:13) α ∗ ( u j − v j ) (cid:13)(cid:13) s, ∞ + (cid:13)(cid:13) u i − v i (cid:13)(cid:13) s, ∞ (cid:13)(cid:13) α ∗ v j (cid:13)(cid:13) s, ∞ (cid:17) ≤ C k α k (cid:16)(cid:13)(cid:13) u i (cid:13)(cid:13) s, ∞ (cid:13)(cid:13) u j − v j (cid:13)(cid:13) s, ∞ + (cid:13)(cid:13) v j (cid:13)(cid:13) s, ∞ (cid:13)(cid:13) u i − v i (cid:13)(cid:13) s, ∞ (cid:17) ≤ k α k P (cid:16) k u k s, ∞ , k v k s, ∞ (cid:17) k u − v k s, ∞ where P is some quadratic polynomial of two variables with nonnegative coefficients.The above results yield the following estimates for u, v ∈ Y ( T ) k Su k X ( T ) ≤ k ϕ k s, ∞ + T k ψ k s, ∞ + C k α k R T , and k Su − Sv k X ( T ) ≤ P ( R, R ) k α k T k u − v k X ( T ) concluding the proofs of (2.3) and (2.5).
4. Global Existence and Blow Up in Finite Time
In this section,we will first show that the maximal time of existence for the solution ofthe Cauchy problem (1.1)-(1.2) depends only on the L ∞ norm of the initial data. Thenwe will prove the existence of a global solution for two classes of nonlinearities and finallyinvestigate blow-up for general nonlinearities.11 .1. Global Existence By repeatedly applying local existence theorems (Theorems 2.2-2.5 and 3.3) the so-lution can be continued to the maximal time interval [0 , T max ) where either T max = ∞ ,i.e. we have a global solution, orlim sup t → T − max ( k u ( t ) k X + k u t ( t ) k X ) = ∞ , where k k X denotes either one of the norms in C b ( R ), L p ( R ) ∩ L ∞ ( R ), C b ( R ), W ,p ( R )or H s ( R ) ∩ L ∞ ( R ). Theorem 4.1.
Assume that the conditions in either one of Theorems 2.2-2.5 or 3.3hold. Then either there is a global solution or maximal time is finite, where T max ischaracterized by the L ∞ blow-up condition lim sup t → T − max k u ( t ) k ∞ = ∞ . Proof.
Clearly in each case the norm k k ∞ is smaller than k k X . Hence it suffices to provethat if lim sup t → T − k u ( t ) k ∞ = M < ∞ , then lim sup t → T − ( k u ( t ) k X + k u t ( t ) k X ) < ∞ .So assume that the solution exists in some interval [0 , T ) and satisfies k u ( t ) k ∞ ≤ R forall 0 ≤ t < T . The solution satisfies u ( x, t ) = ϕ ( x ) + tψ ( x ) + Z t ( t − τ )( Ku )( x, τ ) dτ,u t ( x, t ) = ψ ( x ) + Z t ( Ku )( x, τ ) dτ. In all cases the estimate for Ku is of the form k Ku k X ≤ M ( k u k ∞ ) k u k X with a nondecreasing function M of k u k ∞ . Since k u ( t ) k ∞ ≤ R for all t ∈ [0 , T ), k u ( t ) k X + k u t ( t ) k X ≤ k ϕ k X + (1 + T ) k ψ k X + (1 + T ) M ( R ) Z t k u ( τ ) k X dτ, so that Gronwall’s Lemma implies k u ( t ) k X + k u t ( t ) k X ≤ ( k ϕ k X + (1 + T ) k ψ k X ) e (1+ T ) M ( R ) t for all t ∈ [0 , T ). So lim sup t → T − ( k u ( t ) k X + k u t ( t ) k X ) < ∞ . Theorem 4.2.
Assume that the conditions in either one of Theorems 2.2-2.5 hold. Ifthe nonlinear term w in (1.1) satisfies | w ( η ) | ≤ a | η | + b for all η ∈ R , then there is aglobal solution.Proof. Assume the solution exists on [0 , T ). Then | ( Ku )( x, t ) | ≤ Z R | α ( y − x ) | ( a | u ( y, τ ) − u ( x, τ ) | + b ) dy ≤ a ( | α | ∗ | u | ) ( x, t ) + a k α k | u ( x, t ) | + b k α k , k u ( t ) k ∞ ≤ k ϕ k ∞ + t k ψ k ∞ + Z t ( t − τ ) ( a k ( | α | ∗ | u | ) ( τ ) k ∞ + a k α k k u ( τ ) k ∞ + b k α k ) dτ ≤ k ϕ k ∞ + T k ψ k ∞ + bT k α k + 2 aT k α k Z t k u ( τ ) k ∞ dτ, and Gronwall’s lemma shows that lim sup t → T − k u ( t ) k ∞ < ∞ . Lemma 4.3. (The Energy Identity) Assume that α ∈ L ( R ) is even and w ∈ C ( R ) isodd with w (0) = 0 . If u satisfies the Cauchy problem (1.1)-(1.2) on [0 , T ) with initialdata ϕ, ψ ∈ L ( R ) ∩ L ∞ ( R ) , then the energy E ( t ) = 12 k u t ( t ) k + 12 Z R α ( y − x ) W ( u ( y, t ) − u ( x, t )) dydx, is constant for t ∈ [0 , T ) , where W ( η ) = R η w ( ρ ) dρ .Proof. By Theorem 2.3 with p = 1 we know u ∈ C ([0 , T ] , L ( R ) ∩ L ∞ ( R )). Since L ( R ) ∩ L ∞ ( R ) ⊂ L ( R ), we have u t ( t ) ∈ L ( R ). Moreover, an estimate similar to (2.7)where w is replaced by W shows that the term α ( y − x ) W ( u ( y, t ) − u ( x, t )) is integrable on R . Hence E ( t ) is defined for all t ∈ [0 , T ). Multiplying (1.1) by u t ( x, t ) and integratingin x we obtain Z R u tt ( x ) u t ( x ) dx = Z R α ( y − x ) w ( u ( y ) − u ( x )) u t ( x ) dydx = 12 Z R α ( y − x ) w ( u ( y ) − u ( x )) u t ( x ) dydx + 12 Z R α ( y − x ) w ( u ( y ) − u ( x )) u t ( x ) dydx, where we have again suppressed t . We now change the order of integration and switchthe variables x, y in the last integral to obtain12 Z R α ( x − y ) w ( u ( x ) − u ( y )) u t ( y ) dydx. Since α is even while w is odd, this gives − Z R α ( y − x ) w ( u ( y ) − u ( x )) u t ( y ) dydx, so that Z R u tt ( x ) u t ( x ) dx = − Z R α ( y − x ) w ( u ( y ) − u ( x )) ( u t ( y ) − u t ( x )) dydx. But since W ′ = w ; we have ddt Z R ( u t ( x )) dx = − ddt Z R α ( y − x ) W ( u ( y ) − u ( x )) dydx so that dEdt = 0. 13 heorem 4.4. Assume that α ∈ L ( R ) ∩ L ∞ ( R ) is even with α ≥ almost everywhere; w ∈ C ( R ) is odd with w (0) = 0 and W ≥ . If there is some q ≥ and C > so that | w ( η ) | q ≤ CW ( η ) (4.1) for all η ∈ R , then there is a global solution for initial data ϕ, ψ ∈ L ( R ) ∩ L ∞ ( R ) .Proof. Assume that the solution exists in [0 , T ). By Lemma 4.3 the energy is finite andthe energy identity E ( t ) = E (0) holds for all t ∈ [0 , T ). Consider the energy densityfunction e ( x, t ) = 12 ( u t ( x, t )) + Z R α ( y − x ) W ( u ( y, t ) − u ( x, t )) dy. Differentiating with respect to te t ( x, t ) = u t ( x, t ) u tt ( x, t ) + Z R α ( y − x ) w ( u ( y, t ) − u ( x, t ))( u t ( y, t ) − u t ( x, t )) dy = Z R α ( y − x ) w ( u ( y, t ) − u ( x, t )) u t ( y, t ) dy. Note that by the assumptions of the theorem e ( x, t ) and e t ( x, t ) are in L ∞ ( R ) for eachfixed t . Letting p be the dual index to q ; i.e. 1 /p + 1 /q = 1, we have | e t ( x, t ) | ≤ Z R α ( y − x ) | w ( u ( y, t ) − u ( x, t )) | | u t ( y, t ) | dy ≤ k α k /p ∞ k u t ( t ) k − /p ∞ Z R | u t ( y, t ) | /p ( α ( y − x )) /q | w ( u ( y, t ) − u ( x, t )) | dy, and by H¨older’s inequality | e t ( x, t ) | ≤ k α k /p ∞ k u t ( t ) k − /p ∞ (cid:18)Z R | u t ( y, t ) | dy (cid:19) /p (cid:18)Z R α ( y − x ) | w ( u ( y, t ) − u ( x, t )) | q dy (cid:19) /q . Using the condition (4.1) we have | e t ( x, t ) | ≤ k α k /p ∞ k u t ( t ) k − /p ∞ k u t ( t ) k /p (cid:18) C Z R α ( y − x ) W ( u ( y, t ) − u ( x, t )) dy (cid:19) /q . Since α ≥ W ≥ , by the energy identity we have k u t ( t ) k ≤ E (0). Also, bothterms in e ( x, t ) are nonnegative so that taking essential supremum over x ∈ R , k e t ( t ) k ∞ ≤ k α k /p ∞ (2 E (0)) /p (2 k e ( t ) k ∞ ) / − /p ( C k e ( t ) k ∞ ) /q ≤ C k e ( t ) k r ∞ with r = 1 / − /p + 1 /q = 2 /q − / C in the last line. Notethat when q ≥ / r = 2 /q − / ≤
1. Since e ( x, t ) = e ( x,
0) + Z t e t ( x, τ ) dτ
14e have k e ( t ) k ∞ ≤ k e (0) k ∞ + Z t k e t ( τ ) k ∞ dτ ≤ k e (0) k ∞ + C Z t k e ( τ ) k r ∞ dτ, for all t ∈ [0 , T ). As r ≤
1, we have k e ( t ) k r ∞ ≤ k e ( t ) k ∞ + 1. By Gronwall’s lemma k e ( t ) k ∞ and thus k u t ( t ) k ∞ stay bounded in [0 , T ). Integration again gives k u ( t ) k ∞ ≤ k ϕ k ∞ + Z t k u t ( τ ) k ∞ dτ so that k u ( t ) k ∞ does not blow up in finite time. Remark 4.5.
Considering the typical nonlinearity w ( η ) = | η | ν − η we have W ( η ) = ν +1 | η | ν +1 . Then the exponent q of Theorem 4.4 equals ( ν + 1) /ν and q ≥ if and onlyif ν ≤ . In other words Theorem 4.4 applies to at most cubic nonlinearities.4.2. Blow-up In this section, we will consider the blow-up of the solution for the Cauchy problem(1.1)-(1.2) by the concavity method. For this purpose, we will use the following lemmato prove blow up in finite time.
Lemma 4.6. [15] Suppose H ( t ) , t ≥ is a positive, twice differentiable function satis-fying H ′′ ( t ) H ( t ) − (1 + ν ) ( H ′ ( t )) ≥ where ν > . If H (0) > and H ′ (0) > , then H ( t ) → ∞ as t → t for some t ≤ H (0) /νH ′ (0) . Theorem 4.7.
Suppose that α is even, w is odd, the conditions of Theorem 2.3 hold for p = 1 and α ≥ almost everywhere. If there is some ν > such that ηw ( η ) ≤ ν ) W ( η ) for all η ∈ R , and E (0) = 12 k ψ k + 12 Z R α ( y − x ) W ( ϕ ( y ) − ϕ ( x )) dydx < , then the solution u of the Cauchy problem (1.1)-(1.2) blows up in finite time.Proof. Assume that there is a global solution. Then u ( t ) , u t ( t ) ∈ L ( R ) ∩ L ∞ ( R ) ⊂ L ( R )for all t >
0. Let H ( t ) = k u ( t ) k + b ( t + t ) for some positive constants b and t to bedetermined later. Suppressing the t variable throughout the computations H ′ ( t ) = 2 h u, u t i + 2 b ( t + t ) H ′′ ( t ) = 2 k u t k + 2 h u, u tt i + 2 b. Using (1.1) 2 h u, u tt i = 2 Z R α ( y − x ) w ( u ( y ) − u ( x )) u ( x ) dydx = Z R α ( y − x ) w ( u ( y ) − u ( x )) u ( x ) dydx + Z R α ( y − x ) w ( u ( y ) − u ( x )) u ( x ) dydx. x and y in the second integral and noting that α is even and w is odd we get2 h u, u tt i = Z R α ( y − x ) w ( u ( y ) − u ( x )) u ( x ) dydx − Z R α ( y − x ) w ( u ( y ) − u ( x )) u ( y ) dydx = − Z R α ( y − x ) w ( u ( y ) − u ( x )) ( u ( y ) − u ( x )) dydx. So that 2 h u, u tt i ≥ − ν ) Z R α ( y − x ) W ( u ( y ) − u ( x )) dydx = 4 (1 + 2 ν ) ( 12 k u t k − E (0)) . Hence we get H ′′ ( t ) ≥ ν ) k u t k − ν ) E (0) + 2 b. On the other hand, we have( H ′ ( t )) = 4 [ h u, u t i + b ( t + t )] ≤ k u k k u t k + b ( t + t )] = 4 h k u k k u t k + 2 k u k k u t k b ( t + t ) + b ( t + t ) (cid:17) ] ≤ h k u k k u t k + b k u k + b k u t k ( t + t ) + b ( t + t ) i . Thus H ′′ ( t ) H ( t ) − (1 + ν ) ( H ′ ( t )) ≥ h ν ) k u t k − ν ) E (0) + 2 b i h k u k + b ( t + t ) i − ν ) h k u k k u t k + b k u k + b k u t k ( t + t ) + b ( t + t ) i = [ − ν ) E (0) + 2 b − b (1 + ν )] h k u k + b ( t + t ) i = − ν ) ( b + 2 E (0)) H ( t ) . Now if we choose b ≤ − E (0), this gives H ′′ ( t ) H ( t ) − (1 + ν ) ( H ′ ( t )) ≥ . Moreover H ′ (0) = 2 h ϕ, ψ i + 2 bt > t . According to Lemma 4.6, this implies that H ( t ), and thus k u ( t ) k blows up in finite time contradicting the assumption that the global solution exists. Acknowledgement : This work has been supported by the Scientific and TechnologicalResearch Council of Turkey (TUBITAK) under the project TBAG-110R002.16 eferences [1] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech.Phys. Solid. 48 (2000) 175-209.[2] I. A. Kunin, Elastic Media with Microstructure vol. I and II. Springer, Berlin (1982).[3] D. Rogula, Nonlocal Theory of Material Media, Springer, Berlin (1982).[4] A. C. Eringen, Nonlocal Continuum Field Theories, Springer, New York (2002).[5] S. A. Silling, M. Zimmermann, R. Abeyaratne, Deformation of a peridynamic bar, J. Elasticity 73(2003) 173-190.[6] O. Weckner, R. Abeyaratne, The effect of long-range forces on the dynamics of a bar, J. Mech. Phys.Solid. 53 (2005) 705-728.[7] E. Emmrich, O. Weckner, The peridynamic equation of motion in non-local elasticity theory. InProceeding of III European Conference on Computational Mechanics: Solids, Structures and CoupledProblems in Engineering, C. A. Mota Soares et. al. (eds.), Lisbon, Portugal, (2006).[8] E. Emmrich, O. Weckner, Analysis and numerical approximation of an integro-differential equationmodeling non-local effects in linear elasticity, Math. Mech. Solid. 12 (2007) 363-384.[9] E. Emmrich, O. Weckner, On the well-posedness of the linear peridynamic model and its convergencetowards the Navier equation of linear elasticity, Commun. Math. Sci. 5 (2007) 851-864.[10] Q. Du, K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, M2ANMath. Model. Numer. Anal. 45 (2011) 217-234.[11] N. Duruk, H. A. Erbay, A. Erkip, Global existence and blow-up for a class of nonlocal nonlinearCauchy problems arising in elasticity, Nonlinearity 23 (2010) 107-118.[12] N. Duruk, H. A. Erbay, A. Erkip, Blow-up and global existence for a general class of nonlocalnonlinear coupled wave equations, J. Diff. Eqs. 250 (2011) 1448-1459.[13] H. A. Erbay, S. Erbay, A. Erkip, The Cauchy problem for a class of two-dimensional nonlocalnonlinear wave equations governing anti-plane shear motions in elastic materials, Nonlinearity 24(2011) 1347-1359.[14] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, 1996, pp. 10.[15] V. K. Kalantarov, O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equation ofparabolic and hyperbolic types, J. Soviet Math. 10 (1978) 53-70.[1] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech.Phys. Solid. 48 (2000) 175-209.[2] I. A. Kunin, Elastic Media with Microstructure vol. I and II. Springer, Berlin (1982).[3] D. Rogula, Nonlocal Theory of Material Media, Springer, Berlin (1982).[4] A. C. Eringen, Nonlocal Continuum Field Theories, Springer, New York (2002).[5] S. A. Silling, M. Zimmermann, R. Abeyaratne, Deformation of a peridynamic bar, J. Elasticity 73(2003) 173-190.[6] O. Weckner, R. Abeyaratne, The effect of long-range forces on the dynamics of a bar, J. Mech. Phys.Solid. 53 (2005) 705-728.[7] E. Emmrich, O. Weckner, The peridynamic equation of motion in non-local elasticity theory. InProceeding of III European Conference on Computational Mechanics: Solids, Structures and CoupledProblems in Engineering, C. A. Mota Soares et. al. (eds.), Lisbon, Portugal, (2006).[8] E. Emmrich, O. Weckner, Analysis and numerical approximation of an integro-differential equationmodeling non-local effects in linear elasticity, Math. Mech. Solid. 12 (2007) 363-384.[9] E. Emmrich, O. Weckner, On the well-posedness of the linear peridynamic model and its convergencetowards the Navier equation of linear elasticity, Commun. Math. Sci. 5 (2007) 851-864.[10] Q. Du, K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, M2ANMath. Model. Numer. Anal. 45 (2011) 217-234.[11] N. Duruk, H. A. Erbay, A. Erkip, Global existence and blow-up for a class of nonlocal nonlinearCauchy problems arising in elasticity, Nonlinearity 23 (2010) 107-118.[12] N. Duruk, H. A. Erbay, A. Erkip, Blow-up and global existence for a general class of nonlocalnonlinear coupled wave equations, J. Diff. Eqs. 250 (2011) 1448-1459.[13] H. A. Erbay, S. Erbay, A. Erkip, The Cauchy problem for a class of two-dimensional nonlocalnonlinear wave equations governing anti-plane shear motions in elastic materials, Nonlinearity 24(2011) 1347-1359.[14] M. E. Taylor, Partial Differential Equations III: Nonlinear Equations, Springer, 1996, pp. 10.[15] V. K. Kalantarov, O. A. Ladyzhenskaya, The occurence of collapse for quasilinear equation ofparabolic and hyperbolic types, J. Soviet Math. 10 (1978) 53-70.