Dirichlet boundary conditions for degenerate and singular nonlinear parabolic equations
aa r X i v : . [ m a t h . A P ] D ec DIRICHLET BOUNDARY CONDITIONS FOR DEGENERATEAND SINGULAR NONLINEAR PARABOLIC EQUATIONS
Fabio Punzo , Marta Strani Abstract.
We study existence and uniqueness of solutions to a class of nonlin-ear degenerate parabolic equations, in bounded domains. We show that thereexists a unique solution which satisfies possibly inhomogeneous Dirichlet bound-ary conditions. To this purpose some barrier functions are properly introducedand used.
Keywords.
Parabolic equations, Dirichlet boundary conditions, barrier func-tions, sub– and supersolutions, comparison principle .
AMS subject classification : 35K15, 35K20, 35K55, 35K65, 35K67 .1.
Introduction
We are concerned with bounded solutions to the following nonlinear parabolicequation:(1.1) ρ ∂ t u = ∆[ G ( u )] in Ω × (0 , T ] , where Ω is an open bounded subset of R N ( N ≥
1) with boundary ∂ Ω = S and ρ is apositive function of the space variables. We always make the following assumption: H0. S is an ( N − − dimensional compact submanifold of R N of class C .Moreover, we require the functions ρ , G and f to satisfy the following hypotheses H1. ρ ∈ C (Ω) , ρ > H2. G ∈ C ( R ) , G (0) = 0 , G ′ ( s ) > s ∈ R \ { } . Moreover, if G ′ (0) = 0,then G ′ is decreasing in ( − δ,
0) and increasing in (0 , δ ) for some δ > G and ρ ; to see this, letus think equation (1.1) as(1.2) ∂ t u = 1 ρ ∆[ G ( u )] in Ω × (0 , T ] , and set d ( x ) := dist( x, S ) ( x ∈ ¯Ω) . In fact, in view of the nonlinear function G ( u ) and hypothesis H2 the equation (1.1)can be degenerate ; however, we also consider the case that such a kind of degeneracy Dipartimento di Matematica “F. Enriques”, Universit`a degli Studi di Milano, via C. Saldini 50,20133 Milano, Italy. Email: [email protected]. Dipartimento di Matematica e Applicazioni, Universit`a di Milano Bicocca, via Cozzi 55, 20125Milano, Italy. Email: [email protected], [email protected] . does not occur (see H5 below). Moreover, if the coefficient ρ ( x ) → d ( x ) → ρ ∆ has the coefficient ρ which is unbounded at S , so the operator is singular ; whereas, if ρ ( x ) → ∞ as d ( x ) →
0, the operator ρ ∆ is degenerate at S .Problem (1.1) appears in a wide number of physical applications (see, e.g., [21]);note that, by choosing G ( u ) = | u | m − u for some m >
1, we obtain the well knownporous medium equation with a variable density ρ = ρ ( x ) (see [4, 5]).In the literature, a particular attention has been devoted to the following com-panion Cauchy problem(1.3) ( ρ∂ t u = ∆[ G ( u )] in R N × (0 , T ] ,u = u in R N × { } . In particular, existence and uniqueness of solutions to (1.3) have been extensivelystudied; note that here and hereafter we always consider very weak solutions (seeSection 2.1 for the precise definition). To be specific, if one makes the followingassumptions:( i ) ρ ∈ C ( R N ) , ρ > ii ) u ∈ L ∞ ( R N ) ∩ C ( R N ),it is well known (see [5, 21, 15, 30]) that there exists a bounded solution to (1.3);moreover, for N = 1 and N = 2 such a solution is unique. When N ≥
3, theuniqueness of the solution in the class of bounded functions is no longer guaranteed,and it is strictly related with the behavior at infinity of the density ρ . Indeed, itis possible to prove that if ρ does not decay too fast at infinity, then problem (1.3)admits at most one bounded solution (see [30]). On the contrary, if one supposethat ρ decays sufficiently fast at infinity, then the non uniqueness appears (see[4, 14, 18, 30]).In this direction, in [14] the authors prove the existence and uniqueness of thesolution to (1.3) which satisfies the following additional condition at infinity(1.4) lim | x |→∞ u ( x, t ) = a ( t ) uniformly for t ∈ [0 , T ] , supposing a ∈ C ([0 , T ]) , a > | x |→∞ u ( x ) = a (0) . Note that (1.4) is apoint-wise condition at infinity for the solution u . Also, the results of [14] have beengeneralized in [19, 20] to the case of more general operators.When considering equation (1.1) in a bounded subset Ω ⊂ R N , in view of H1 ,since ρ is allowed either to vanish or to diverge at S , it is natural to consider thefollowing initial value problem associated with (1.1):(1.5) ( ρ∂ t u = ∆[ G ( u )] in Ω × (0 , T ] ,u = u in Ω × { } , where no boundary conditions are specified at S . We require ρ , G and f to satisfyhypotheses H1-2-3 ; furthermore, for the initial datum u we assume that IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 3
H3. u ∈ L ∞ (Ω) ∩ C (Ω) . Concerning the existence and uniqueness of the solutions to (1.5), the case G ( u ) = u has been largely investigated, using both analytical and stochastic methods (see,e.g., [23, 28, 29, 32]). Also analogous elliptic or elliptic-parabolic equations haveattracted much attention in the literature (see, e.g., [6, 7, 8, 9, 10, 11, 26, 27]); inparticular, the question of prescribing continuous data at S has been addressed (see,e.g., [23, 27, 28, 29]).For general nonlinear function G , the well-posedness of problem (1.5) has beenstudied in [17] in the case N = 1 and subsequently addressed for N ≥ ρ diverges sufficiently fast as d ( x ) →
0, thenone has uniqueness of bounded solutions not satisfying any additional condition at S . Indeed, if one requires that there exists ˆ ε > ρ ∈ C ((0 , ˆ ε ]) such that • ρ ( x ) ≥ ρ ( d ( x )) >
0, for any x ∈ S ˆ ε := { x ∈ Ω | d ( x ) < ˆ ε } , • R ˆ ε η ρ ( η ) dη = + ∞ ,then there exists at most one bounded solution to (1.5).Conversely, if either ρ ( x ) → ∞ sufficiently slow or ρ does not diverge when d ( x ) → ε > ρ ∈ C ((0 , ˆ ε ]) such that • ρ ( x ) ≤ ρ ( d ( x )), for any x ∈ S ˆ ε , • R ˆ ε η ρ ( η ) dη < + ∞ ,then, for any A ∈ Lip([0 , T ]), A (0) = 0, there exists a solution to (1.5) satisfying(1.6) lim d ( x ) → | U ( x, t ) − A ( t ) | = 0 , uniformly with respect to t ∈ [0 , T ], where U is defined as U ( x, t ) := Z t G ( u ( x, τ )) dτ. In particular, the previous result implies non-uniqueness of bounded solutions to(1.5). Moreover, the solution to problem (1.5) which satisfies (1.6) is unique, pro-vided A ≡ G ( u ) = u. Formally, the boundary S for problem (1.5) plays the same role played by infinity for the Cauchy problem (1.3); hence, the well-posedness for (1.5) depends on thebehavior of ρ in the limit d ( x ) →
0, in analogy with the previous results for theCauchy problem (1.3), where it depends on the behavior of ρ for large | x | .Thus, a natural question that arises is if it is possible to impose at S Dirichletboundary conditions, instead of the integral one (1.6). Moreover, on can ask ifsuch a Dirichlet condition restore uniqueness in more general situations than theones considered in connection with (1.6). Observe that, as recalled above, the samequestion has already been investigated for the linear case G ( u ) = u (see, e.g., [23,27, 28, 29]), and for the case that ρ ≡ G is general (see [2, 3]). The case F. PUNZO, M.STRANI where both ρ and G are general, which is a quite natural situation also for variousapplications (see, e.g., [22]), has not been treated in the literature and is the objectof our investigation.In fact, the main novelty of our paper relies in the following result: we proveexistence and uniqueness of a bounded solution to problem (1.5) satisfying Dirichletpossibly non-homogeneous boundary conditions. This is of course a much strongercondition with respect to (1.6). As in [31], we require the function ρ to satisfy H4. there exists ˆ ε > ρ ∈ C ((0 , ˆ ε ]) such that i . ρ ( x ) ≤ ρ ( d ( x )), for any x ∈ S ˆ ε , ii . R ˆ ε η ρ ( η ) dη < + ∞ .A natural choice for ρ is given by(1.7) ρ ( η ) = η − α , for some α ∈ ( −∞ , , and η ∈ (0 , ˆ ε ] . Under the hypothesis H4 , we show that, for any ϕ ∈ C ( S × [0 , T ]) , if either G isnon degenerate, i.e. there holds H5. G ∈ C ( R ) , G ′ ( s ) ≥ α > s ∈ R , or ϕ and u satisfy(1.8) ϕ > S × [0 , T ] , lim inf x → x u ( x ) ≥ α > x ∈ S , then there exists a unique bounded solution to (1.5) such that, for each τ ∈ (0 , T ),(1.9) lim x → x t → t u ( x, t ) = ϕ ( x , t ) uniformly with respect to t ∈ [ τ, T ] and x ∈ S . If we drop either the assumption of non-degeneracy on G or the assumption (1.8),we need to restrict our analysis to the special class of data ϕ which only depend on x ; in fact, for any ϕ ∈ C ( S ) we prove that there exists a unique bounded solutionto (1.5) satisfying, for each τ ∈ (0 , T ),(1.10) lim x → x u ( x, t ) = ϕ ( x ) uniformly with respect to t ∈ [ τ, T ] and x ∈ S , provided(1.11) lim x → x u ( x ) = ϕ ( x ) for every x ∈ S . To prove the existence results we introduce and use suitable barrier functions(see (3.14), (3.21), (3.27), (3.32), (3.36), (3.43), (3.44), (3.45) below). We shouldnote that the definitions of such barriers seem to be new. Let us observe thatin constructing such barrier functions, always supposing that H4 holds, the casesinf Ω ρ > ρ ∈ L ∞ (Ω) will be treated separately (for more details, see Section3). To explain the differences among these two cases, let us refer to the model case,in which hypothesis H4 holds with ρ given by (1.7). So, the previous two casescorrespond to the choices α < α ∈ [0 , ρ ∆ has a prominent role. From this viewpointwe can say that the previous two cases are deeply different, since, when α ∈ (0 , IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 5 the operator ρ ∆ is degenerate at S , whereas, when α <
0, it is singular, in the sensethat its coefficient ρ blows-up at S . Clearly, the choice α = 0 recasts in both cases.In constructing our barrier functions, besides taking into account the behavior at S of the density ρ ( x ) as described above, we have to overcome some difficulties dueto the nonlinear function G ( u ). In this respect, we should note that on the one hand,barrier functions similar to those we construct were used in [14] and in [19], whereproblem (1.3) was addressed and conditions were prescribed at infinity. However,such barriers cannot be trivially adapted to our case. Indeed, by an easy variationof them we could only consider S in place of infinity , prescribing u ( x, t ) → a ( t ) as d ( x ) → t ∈ (0 , T ]), but we cannot distinguish different points x ∈ S and imposeconditions (1.9) and (1.10) . On the other hand, other similar barriers were used inthe literature (see, e.g., [12]) to prescribe Dirichlet boundary conditions to solutionsto linear parabolic equations, in bounded domains; similar results have also beenestablished for linear elliptic equations (see [13], [25]); however, they cannot be usedin our situation, in view of the presence of the nonlinear function G ( u ).Let us mention that our results have some connections with regularity results upto the boundary. In fact, as a consequence of our existence and uniqueness results,any solution to problem (1.5) is continuous in Ω × [0 , T ] . Similar regularity resultscould be deduced from results in [2] and in [3], where more general equations aretreated, only when(1.12) C ≤ ρ ( x ) ≤ C for all x ∈ Ω , for some 0 < C < C . However, we suppose hypotheses H1 and H5 , that areweaker than (1.12) .We close this introduction with a brief overview of the paper. In Section 2 wepresent a description of the main contributions of the paper; in particular, we stateTheorem 2.3, Theorem 2.4 and Theorem 2.5, that assure, under suitable hypotheses,the existence of a bounded solution to (1.5) satisfying a proper Dirichlet boundarycondition. Subsequently, we show that such a solution is unique (see Theorem 2.7).Section 3 is devoted to the proofs of the existence results, while in Section 4 theproof of the uniqueness result is given.2. Statement of the main results
In this section we present existence and uniqueness results for bounded solutionsto(2.1) ( ρ∂ t u = ∆[ G ( u )] in Ω × (0 , T ] ,u = u in Ω × { } , where Ω ⊂ R N satisfies hypothesis H0 , and ρ , G and u satisfy hypotheses H1-4 .In the following, we will extensively use the following notations: • Q T := Ω × (0 , T ]; • S ε := { x ∈ Ω : d ( x ) < ε } ( ε > F. PUNZO, M.STRANI • A ε := ∂ S ε ∩ Ω; • Ω ε := Ω \ S ε .2.1. Mathematical background.
Before stating our results, let us define the toolswe shall use in the following.
Definition 2.1.
A function u ∈ C (Ω × [0 , T ]) ∩ L ∞ (Ω × (0 , T )) is a solution to (2.1) if (2.2) Z τ Z Ω (cid:2) u ρ ∂ t ψ + G ( u )∆ ψ (cid:3) dx dt == Z Ω (cid:2) u ( x, T ) ψ ( x, T ) − u ( x ) ψ ( x, (cid:3) ρ ( x ) dx + Z τ Z ∂ Ω G ( u ) h∇ ψ, ν i dS dt, for any open set Ω with smooth boundary ∂ Ω such that Ω ⊂ Ω , for any τ ∈ (0 , T ] and for any ψ ∈ C , x,t (Ω × [0 , τ ]) , ψ ≥ , ψ = 0 in ∂ Ω × [0 , τ ] , where ν denotes theouter normal to Ω .Moreover, we say that u is a supersolution (subsolution respectively) to (2.1) if (2.2) holds with ≤ ( ≥ respectively ). Given ε >
0, we also consider the following auxiliary problem(2.3) ρ∂ t u = ∆[ G ( u )] in Ω ε × (0 , T ] := Q εT ,u = φ in A ε × (0 , T ) ,u = u in Ω ε × { } ;where φ ∈ C ( A ε × [0 , T ]) , φ ( x,
0) = u ( x ) for all x ∈ A ε . Definition 2.2.
A function u ∈ C (Ω ε × [0 , T ]) is a solution to (2.1) if Z τ Z Ω (cid:2) u ρ ∂ t ψ + G ( u )∆ ψ (cid:3) dx dt = Z Ω (cid:2) u ( x, T ) ψ ( x, T ) − u ( x ) ψ ( x, (cid:3) ρ ( x ) dx + Z τ Z ∂ Ω \A ε G ( u ) h∇ ψ, ν i dS dt + Z τ Z ∂ Ω ∩A ε G ( φ ) h∇ ψ, ν i dS dt, (2.4) for any open set Ω ⊂ Ω ε with smooth boundary ∂ Ω , for any τ ∈ (0 , T ] and forany ψ ∈ C , x,t (Ω × [0 , τ ]) , ψ ≥ , ψ = 0 in ∂ Ω × [0 , τ ] , where ν denotes the outernormal to Ω . Supersolution and subsolution are defined accordingly. Existence results.
At first, we consider the case of nondegenerate nonlinari-ties G satisfying hypothesis H5 .. IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 7
Theorem 2.3.
Let hypotheses
H0-H1 , H3-H5 be satisfied. Let ϕ ∈ C ( S × [0 , T ]) .Then there exists a maximal solution to (2.1) such that, for each τ ∈ (0 , T ) , (2.5) lim x → x t → t u ( x, t ) = ϕ ( x , t ) , uniformly with respect to t ∈ [ τ, T ] and x ∈ S . We can also prove similar results to Theorem 2.3 in the case of a general nonlin-earity G satisfying H2 . Theorem 2.4.
Let hypotheses
H0-4 be satisfied and let ϕ ∈ C ( S ) . Suppose thatcondition (1.11) holds. Then there exists a maximal solution to (2.1) such that (2.6) lim x → x u ( x, t ) = ϕ ( x ) , uniformly with respect to t ∈ [0 , T ] and x ∈ S . Finally, we can also consider data ϕ and u satisfying(2.7) ϕ > S × [0 , T ] and lim inf x → x u ( x ) ≥ α > x ∈ S . Theorem 2.5.
Let hypothesis
H0-4 be satisfied and let ϕ ∈ C ( S × [0 , T ]) . Supposethat (2.7) holds. Then there exists a maximal solution to (2.1) such that (2.5) holds. Remark 2.6.
If we further suppose that(2.8) lim x → x u ( x ) = ϕ ( x ,
0) for every x ∈ S , then in Theorems 2.3 and 2.5 we can take τ = 0.2.3. Uniqueness results.Theorem 2.7.
Let hypotheses
H0-4 be satisfied, and let ϕ ∈ C ( S × [0 , T ]) . Supposethat (2.7) holds. Then there exists at most one bounded solution to (2.1) such that (2.5) holds. Remark 2.8.
If we consider either the case of a non-degenerate nonlinearity G satisfying H5 , or if we require that ϕ ( x, t ) = ϕ ( x ) for all t ∈ (0 , T ], the previousuniqueness result still holds. It can be shown by using the same arguments as inTheorem 2.7. 3. Existence results: proofs
Preliminaries.
In the proofs of our existence results, in order to show thatthe solution we construct is maximal , we will make use of the following lemma.
Lemma 3.1.
Let hypotheses
H0-4 be satisfied. Let u be a subsolution to problem (2.1) and let ˆ u be a supersolution to problem (2.1) . Suppose that for each τ ∈ (0 , T ) there exists ε τ > such that, for all < ε < ε τ , (3.1) u ≤ ˆ u in A ε × ( τ, T ] . Then u ≤ ˆ u in Q T . F. PUNZO, M.STRANI
In order to prove Lemma 3.1, we need to state the following result; for its proof, see[1, Lemma 10].
Lemma 3.2.
Let ε > . Let (3.2) a := (cid:26) [ G ( u ) − G (ˆ u )] / ( u − ˆ u ) for u = ˆ u, , with u and ˆ u as in Lemma 3.1. Then there exists a sequence { a n } ∈ C ∞ ( Q εT ) suchthat n N +1 ≤ a n ≤ k a k L ∞ ( Q εT ) + 1 n N +1 and ( a n − a ) √ a n → L ( Q εT ) . Furthermore, let χ ∈ C ∞ (Ω ε ) with ≤ χ ≤ . Then there exists a unique solution ψ n ∈ C , x,t ( Q εT ) to problem (3.3) ( ρ∂ t ψ n + a n ∆ ψ n = 0 in Q εT ,ϕ n ( x, T ) = χ ( x ) in Ω ε . Moreover, ψ n has the following properties: i. ≤ ψ n ≤ on Q εT ; ii. R R Q εT a n | ∆ ψ n | < C , for some C > independent of n . iii. sup ≤ t ≤ T R Ω ε |∇ ψ n | < C , for some C > independent of n . Proof of Lemma 3.1.
The proof of this lemma is an adaptation of the argumentsused in [1, Proposition 9]. Let a be as in (3.2); since u and ˆ u are respectivelysubsolution and supersolution to (2.1), in view of the Definition 2.1, with Ω and ψ as in Definition 2.2, by (2.4) with τ = T , we get Z Ω ε ρ ( x )[ u ( x, T ) − ˆ u ( x, T )] ψ ( x, T ) dx − Z T Z Ω ε ( u − ˆ u ) { ∂ t ψ + a ∆ ψ } dt dx ≤− Z τ Z A ε [ G ( u ) − G (ˆ u )] h∇ ψ, ν i dSdt − Z Tτ Z A ε [ G ( u ) − G (ˆ u )] h∇ ψ, ν i dSdt. (3.4)Now, let { a n } and ψ n as in Lemma 3.2. Since, for every n ∈ N , there holds h∇ ψ n , ν i ≤ A ε , if we set ψ = ψ n in (3.4), using (3.1), we obtain Z Ω ε ρ [ u ( x, T ) − ˆ u ( x, T )] χ ( x ) dx − Z T Z Ω ε ( u − ˆ u )( a − a n ) ∆ ψ n dt dx ≤− Z τ Z A ε [ G ( u ) − G (ˆ u )] h∇ ψ n , ν i dS dt − Z Tτ Z A ε [ G ( u ) − G (ˆ u )] h∇ ψ n , ν i dS dt ≤≤ − Z τ Z A ε [ G ( u ) − G (ˆ u )] h∇ ψ n , ν i dS dt. (3.5) IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 9
In view of Lemma 3.2, we get (cid:12)(cid:12)(cid:12)(cid:12)Z T Z Ω ε ( u − ˆ u )( a − a n ) ∆ ψ n dt dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) a − a n √ a n (cid:13)(cid:13)(cid:13)(cid:13) L ( Q T ) k√ a n ∆ ψ n k L ( Q T ) ≤ C √ C (cid:13)(cid:13)(cid:13)(cid:13) a − a n √ a n (cid:13)(cid:13)(cid:13)(cid:13) L ( Q T ) → n → ∞ , (3.6)where the constant C > k u k L ∞ and k ˆ u k L ∞ . Furthermore, (cid:12)(cid:12)(cid:12) Z τ Z A ε [ G ( u ) − G (ˆ u )] h∇ ψ n , ν i dS dt (cid:12)(cid:12)(cid:12) ≤≤ (cid:18)Z τ Z Ω ε [ G ( u ) − G (ˆ u )] dx dt (cid:19) (cid:18)Z τ Z Ω ε |∇ ψ n | dx dt (cid:19) ≤≤ C (cid:18)Z τ Z Ω ε |∇ ψ n | dx dt (cid:19) ≤ C τ √ C, (3.7)where we used Lemma 3.2, ( iii ). Hence, in view of (3.6) and (3.7), letting n → ∞ in (3.5) and then τ →
0, we end up with(3.8) Z Ω ε ρ ( x )[ u ( x, T ) − ˆ u ( x, T )] χ ( x ) dx ≤ . Since (3.8) holds for every χ ∈ C ∞ (Ω ε ), by approximation it also holds with χ ( x ) =sign( u ( x, T ) − ˆ u ( x, T )) + , x ∈ Ω ε . This implies u ≤ ˆ u in Q εT , from which the thesisimmediately follows, letting ε → + . (cid:3) Proofs of the Theorems.
In view of the assumption on ρ ( x ) given in H4 ,there holds the following lemma (see [31]). Lemma 3.3.
Let hypotheses
H0- H4 be satisfied. Then there exists a function V ( x ) ∈ C ( S ε ) such that ∆ V ( x ) ≤ − ρ ( x ) , for all x ∈ S ε ,V ( x ) > , for all x ∈ S ε ,V ( x ) → d ( x ) → . In this section we use the fact that for any ϕ ∈ C ( S × [0 , T ]), there exists(3.9) ˜ ϕ ∈ C ( Q T ) such that ˜ ϕ = ϕ in S × [0 , T ] . We shall write ˜ ϕ ≡ ϕ . Proof of Theorem 2.3.
The proof is divided into two main parts. At first, weconsider that case of a density ρ satisfying hypothesis H4 andinf Ω ρ > . Let η >
0. For any 0 < η < η , we define u ηε ∈ C (Ω ε × [0 , T ]) as the unique solution(see [24]) to(3.10) ρ ∂ t u = ∆ (cid:2) G ( u ) (cid:3) in Ω ε × (0 , T ) ,u = ϕ + η on A ε × (0 , T ) ,u = u ,ε + η in Ω ε × { } , where u ,ε ( x ) := ζ ε u ( x ) + (1 − ζ ε ) ϕ ( x,
0) in Ω ε , and { ζ ε } ⊂ C ∞ c (Ω ε ) is a sequence of functions such that, for any ε >
0, 0 ≤ ζ ε ≤ ζ ε ≡ ε . By the comparison principle, there holds(3.11) | u ηε | ≤ K := max {k u k ∞ , k ϕ k ∞ } + η in Ω ε × (0 , T ) . Moreover, by usual compactness arguments (see, e.g., [24]), there exists a subse-quence { u ηε k } ⊆ { u ηε } which converges, as ε k →
0, locally uniformly in Ω × [0 , T ], toa solution u η to the following problem(3.12) ρ ∂ t u = ∆ (cid:2) G ( u ) (cid:3) in Ω × (0 , T ] ,u = u + η in Ω × { } . We want to prove that, for each τ ∈ (0 , T ),lim x → x t → t u η ( x, t ) = ϕ ( x , t ) , uniformly with respect to t ∈ ( τ, T ] , x ∈ S and η ∈ (0 , η ).Take any τ ∈ (0 , T / . Let ( x , t ) ∈ S × [2 τ, T ]. Set N εδ ( x ) := B δ ( x ) ∩ Ω ε forany δ > ε > ϕ and since G ∈ C ( R ) is increasing, there follows that, for any σ >
0, there exists δ ( σ ) > x , t ) , such that(3.13) G − (cid:2) G ( ϕ ( x , t ) + η ) − σ (cid:3) ≤ ϕ ( x, t ) + η ≤ G − (cid:2) G ( ϕ ( x , t ) + η ) + σ (cid:3) , for all ( x, t ) ∈ N δ ( x ) × ( t δ , t δ ), where t δ := t − δ , and t δ := min { t + δ, T } , and N δ ( x ) := B δ ( x ) ∩ Ω . Clearly, t δ > τ . Now, for any ( x, t ) ∈ N δ ( x ) × ( t δ , t δ ), we define(3.14) w ( x, t ) := G − (cid:2) − M V ( x ) − σ + G ( ϕ ( x , t ) + η ) − λ ( t − t ) − β | x − x | (cid:3) , with V ( x ) as in Lemma 3.3 and M , λ and β positive constants to be fixed conve-niently in the sequel.First of all we want to prove that(3.15) ρ∂ t w ≤ ∆ G ( w ) in N εδ ( x ) × ( t δ , t δ ) . IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 11
To his purpose, we note that ρ∂ t w ≤ ρ λδα , and ∆ G ( w ) ≥ M ρ − βN. Hence, the function w solves (3.15), if(3.16) M ≥ βN inf Ω ρ + 2 λδα . Going further, for any ( x, t ) ∈ [ B δ ( x ) ∩ A ε ] × ( t δ , t δ ), we have(3.17) w ( x, t ) ≤ G − [ G ( ϕ ( x , t ) + η ) − σ ] . Moreover, for ( x, t ) ∈ [ ∂B δ ( x ) ∩ Ω ε ] × ( t δ , t δ ), there holds(3.18) w ( x, t ) ≤ − K, provided β ≥ G ( || ϕ || L ∞ + η ) − G ( − K ) δ . Finally, for all ( x, t ) ∈ N εδ ( x ) × { t δ } , there holds(3.19) w ( x, t ) ≤ G − [ G ( ϕ ( x , t ) + η ) − λδ ] ≤ − K, assuming λ ≥ G ( || ϕ || L ∞ + η ) − G ( − K ) δ . From (3.17), (3.18) and (3.19) we obtain that w is a subsolution to the followingproblem(3.20) ρ ∂ t u = ∆ (cid:2) G ( u ) (cid:3) in N εδ ( x ) × ( t δ , t δ ) ,u = − K in [ ∂B δ ( x ) ∩ Ω ε ] × ( t δ , t δ ) ,u = G − [ G ( ϕ + η ) − σ ] in [ B δ ( x ) ∩ A ε ] × ( t δ , t δ ) ,u = − K in N εδ ( x ) × { t δ } . Recalling the definition of u ηε given in (3.10), and by using (3.11), it follows that u η is a supersolution to problem (3.20). Note that sub– and supersolutions to problem(3.20) are meant similarly to Definition 2.2, considering that N εδ ( x ) is piece-wisesmooth; the same holds for problems of the same form we mention in the sequel.By proceeding with the same methods, for all ( x, t ) ∈ N δ ( x ) × ( t δ , t δ ) we define(3.21) w ( x, t ) := G − (cid:2) M V ( x ) + σ + G ( ϕ ( x , t ) + η ) + λ ( t − t ) + β | x − x | (cid:3) , proving that, with an appropriate choice for the coefficients M , λ and β , w is asupersolution to problem(3.22) ρ ∂ t u = ∆ (cid:2) G ( u ) (cid:3) in N εδ ( x ) × ( t δ , t δ ) ,u = K in [ ∂B δ ( x ) ∩ Ω ε ] × ( t δ , t δ ) ,u = G − [ G ( ϕ + η ) + σ ] in [ B δ ( x ) ∩ ∂ Ω ε ] × ( t δ , t δ ) ,u = K in N εδ ( x ) × { t δ } . Precisely, we require M to be such that M ≥ βN inf Ω ρ + 2 λδα , while β and λ are chosen so that β ≥ G ( K ) − G ( || ϕ || L ∞ + η ) δ , λ ≥ G ( K ) − G ( || ϕ || L ∞ + η ) δ . On the other hand, u η is a subsolution to problem (3.22). Hence, by the comparisonprinciple, and by letting ε k →
0, we get(3.23) w ≤ u η ≤ w in N δ ( x ) × ( t δ , t δ ) . Take any τ ∈ (0 , T /
2) and ( x , t ) ∈ S× [2 τ, T ]. Due to (3.23), recalling the definitionof w and w and by letting x → x , t → t , one has G − (cid:2) G ( ϕ ( x , t ) + η ) − σ (cid:3) ≤ u η ( x , t ) ≤ G − (cid:2) G ( ϕ ( x , t ) + η ) + 2 σ (cid:3) . Letting σ → + , we end up withlim x → x t → t u η ( x, t ) = ϕ ( x , t ) , uniformly with respect to t ∈ (2 τ, T ), x ∈ S and η ∈ (0 , η ), for each τ ∈ (0 , T / { u η k } ⊂ { u η } which converges, as η k →
0, to a solution u to (2.1), locally uniformly in Ω × [0 , T ].Hence, by using (3.23), we have, in the limit σ → + and η → + ,lim x → x t → t u ( x, t ) = ϕ ( x , t ) , uniformly with respect to t ∈ (2 τ, T ) and x ∈ S , for each τ ∈ (0 , T / u is the maximal solution. To this end, let v be anysolution to problem (2.1) satisfying (2.5). From (3.23) it follows that for any α ∈ (0 , η /
4) and for any τ ∈ (0 , T ), there exists ˜ ε > < ε < ˜ ε and η ∈ (0 , η )(3.24) v ( x, t ) ≤ ϕ ( x, t ) + α ≤ u η ( x, t ) for all ( x, t ) ∈ A ε × ( τ, T ] . Moreover(3.25) v ( x,
0) = u ( x ) < u ( x ) + η = u η ( x,
0) for all x ∈ Ω . IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 13
Since v ( x, t ) and u η ( x, t ) are solutions to the same equations in Ω × (0 , T ], in viewof (3.24), (3.25) and Lemma 3.1 there holds v ( x, t ) ≤ u η ( x, t ) for all ( x, t ) ∈ Q T . Passing to the limit η → + we obtain v ≤ u in Q T , and the proof is complete, in this case.In the second part of the proof, we consider a density ρ such that ρ ∈ L ∞ (Ω).Now, we need to slightly modify the arguments used above. Since S ∈ C , by[13] the uniform exterior sphere condition is satisfied, i.e. there exists R > x ∈ S we can find x ∈ R N \ ¯Ω such that B ( x , R ) ⊂ R N \ ¯Ω and B ( x , R ) ∩ S = { x } . Thus, by standard arguments (see K. Miller [25]), it is proventhat the following function(3.26) h ( x ) := C [ e − a R − e − a | x − x | ]satisfies • ∆ h ≤ − B R ( x ); • h > x ∈ (cid:2) ¯ B R ( x ) ∩ ¯Ω (cid:3) \ { x } ; • h ( x ) = 0,for a suitable choice of the constants C > a >
0, independent of x ∈ S .The function h ( x ) can be used in order to built suitable barrier functions w ( x, t )and w ( x, t ). To this end, for ( x, t ) ∈ N δ ( x ) × ( t δ , t δ ), we define(3.27) w ( x, t ) := G − (cid:2) − M h ( x ) − σ + G ( ϕ ( x , t ) + η ) − λ ( t − t ) (cid:3) , being h ( x ) as in (3.26).First of all, because of the properties of h ( x ), there holds ρ∂ t w ≤ ∆ G ( w ), if M ≥ ρ ( x ) λ δα , Hence, we require that M ≥ λ δα k ρ k L ∞ . Next, let ( x, t ) ∈ [ B δ ( x ) ∩ A ε ] × ( t δ , t δ ); we have(3.28) w ≤ G − [ G ( ϕ ( x , t ) + η ) − σ ] . Moreover, for ( x, t ) ∈ [ ∂B δ ( x ) ∩ Ω ε ] × ( t δ , t δ ) we have(3.29) w ( x, t ) ≤ − K, provided M ≥ G ( || ϕ || L ∞ + η ) − G ( − K )inf ∂B δ ( x ) ∩ Ω h . Finally, for ( x, t ) ∈ N εδ ( x ) × { t δ } (3.30) w ( x, t ) ≤ G − [ G ( ϕ ( x , t ) + η ) − λδ ] ≤ − K imposing λ ≥ G ( || ϕ || L ∞ + η ) − G ( − K ) δ . From (3.28), (3.29) and (3.30) we can state that w is a subsolution to the followingproblem(3.31) ρ ∂ t u = ∆ (cid:2) G ( u ) (cid:3) in N ε,ε × ( t δ , t δ ) ,u = − K on [ ∂B δ ( x ) ∩ Ω ε ] × ( t δ , t δ ) ,u = G − [ G ( ϕ + η ) − σ ] in [ B δ ( x ) ∩ ∂ Ω ε ] × ( t δ , t δ ) ,u = − K in N εδ ( x ) × { t δ } , while u η is a supersolution to the same problem. By proceeding with the samemethods, for all ( x, t ) ∈ N δ ( x ) × ( t δ , t δ ) we define(3.32) w ( x, t ) := G − (cid:2) M h ( x ) + σ + G ( ϕ ( x , t ) + η ) + λ ( t − t ) (cid:3) , proving that, with the appropriate choices for the coefficients M , λ and β , w is asuper-solution to problem(3.33) ρ ∂ t u = ∆ (cid:2) G ( u ) (cid:3) in N ε,ε × ( t δ , t δ ) ,u = K on [ ∂B δ ( x ) ∩ Ω ε ] × ( t δ , t δ ) ,u = G − [ G ( ϕ + η ) + σ ] in [ B δ ( x ) ∩ ∂ Ω ε ] × ( t δ , t δ ) ,u = K in N εδ ( x ) × { t δ } , while u η is a subsolution to the same problem. Hence, by the comparison principle,and by letting ε k →
0, we get(3.34) w ≤ u η ≤ w in N δ ( x ) × ( t δ , t δ ) . Take any τ ∈ (0 , T / . Let ( x , t ) ∈ S × [2 τ, T ]. In view of (3.34), recalling thedefinition of w and w and by letting x → x and choosing t = t , one has G − (cid:2) G ( ϕ ( x , t ) + η ) − σ (cid:3) ≤ u η ( x, t ) ≤ G − (cid:2) G ( ϕ ( x , t ) + η ) + 2 σ (cid:3) . So, the thesis follows for σ → + as in the previous case, as well as the maximalityof u . (cid:3) Proof of Theorem 2.4.
As in the proof of Theorem 2.3, we consider at first thecase of a density ρ satisfying hypothesis H4 and inf Ω ρ > IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 15
We define u ηε ∈ C (Ω ε × [0 , T ]) as the unique solution to (3.10). Take any x ∈ S .Observe that from (1.11) we can infer that for any σ > δ = δ ( σ ) > x , such that(3.35) G − (cid:2) G ( ϕ ( x ) + η ) − σ (cid:3) ≤ u ( x ) + η ≤ G − (cid:2) G ( ϕ ( x ) + η ) + σ (cid:3) for all x ∈ N δ ( x ) . For all x ∈ N δ ( x ), we define(3.36) w ( x ) := G − (cid:2) − M V ( x ) − σ + G ( ϕ ( x ) + η ) − β | x − x | (cid:3) , where V is defined in Lemma 3.3, and M and β are positive constants to be chosen.There holds ∆ G ( w ) ≥ M ρ − βN ≥ , provided(3.37) M ≥ βN inf Ω ρ . Going further, for all ( x, t ) ∈ [ B δ ( x ) ∩ A ε ] × (0 , T ), there holds(3.38) w ≤ ϕ ( x ) + η, while, for all ( x, t ) ∈ [ ∂B δ ( x ) ∩ Ω ε ] × (0 , T ) w ≤ − K, provided β ≥ G ( | ϕ ( x ) | ) − G ( − K ) δ . Moreover, from (3.35) it follows that(3.39) w ( x ) ≤ u ( x ) + η for all x ∈ N εδ ( x ) . Thus w is a subsolution, while u η is a supersolution to problem(3.40) ρ ∂ t u = ∆ (cid:2) G ( u ) (cid:3) in N εδ ( x ) × (0 , T ) ,u = − K on [ ∂B δ ( x ) ∩ Ω ε ] × (0 , T ) ,u = G − [ G ( ϕ + η ) − σ ] in [ B δ ( x ) ∩ ∂ Ω ε ] × (0 , T ) ,u = u + η in N εδ ( x ) × { } , By the comparison principle, there holds(3.41) w ≤ u ηε in N εδ ( x ) × (0 , T ) . Analogously, we have(3.42) u ηε ≤ w in N εδ ( x ) × (0 , T ) , where(3.43) w ( x ) := G − (cid:2) M V ( x ) + σ + G ( ϕ ( x ) + η ) (cid:3) , with M >
From (3.41) and (3.42) with ε = ε k →
0, we obtain w ≤ u η ≤ w in N δ ( x ) × (0 , T ) , where u η is a solution to problem (3.12). Hence the thesis follows by letting x → x and σ → + , as in the proof of Theorem 2.3.By slightly modifying the previous arguments, it is possible to prove Theorem2.4 also in the case of a density ρ satisfying ρ ∈ L ∞ (Ω). Indeed, we construct thebarrier functions w ( x ) and w ( x ) as(3.44) w ( x ) := G − (cid:2) − M h ( x ) − σ + G ( ϕ ( x ) + η ) − β | x − x | (cid:3) , (3.45) w ( x ) := G − (cid:2) M h ( x ) + σ + G ( ϕ ( x ) + η ) + β | x − x | (cid:3) , being h ( x ) as in (3.26). The thesis follows as in the second part of the proof ofTheorem 2.3, by making use of the properties of h ( x ) and by suitable choices of theconstants M , β, M , β . (cid:3) Proof of Theorem 2.5.
Let α := min n min ¯Ω × [0 ,T ] ϕ, α o , with α > ϕ ∈ C ( S × [0 , T ]) and ϕ > S × [0 , T ], we canselect ˜ ϕ ≡ ϕ as in (3.9), such that ˜ ϕ > Q T . So, α > . Take u ∈ C ( ¯Ω) suchthat(3.46) u ≤ u in Ω , lim x → x u ( x ) = α . By Theorem 2.4, there exists a solution u ( x, t ) to the following problem(3.47) ( ρ∂ t u = ∆[ G ( u )] in Ω × (0 , T ] ,u = u in Ω × { } , such that(3.48) lim x → x u ( x, t ) = α x ∈ S , t ∈ [0 , T ] . We construct the approximating sequence { u ηε } as in the proof of Theorem 2.3. Dueto (3.47) and (3.48), by the comparison principle, we have that for some ε >
0, forevery 0 < ε < ε (3.49) u ( x, t ) ≤ u ηε ( x, t ) for all x ∈ Ω ε , t ∈ (0 , T ] . Then there exists a subsequence { u ηε k } ⊂ { u ηε } which converges, as ε k →
0, to asolution u η to (3.12). From (3.49) it follows that u η ( x, t ) ≥ u ( x, t ) for all x ∈ Ω , t ∈ (0 , T ] . Therefore, for some 0 < ε < ε , for all 0 < η < η there holds(3.50) u η ( x, t ) ≥ α x ∈ S ε , t ∈ (0 , T ] . Hence, in S ε × (0 , T ] the equation does not degenerate, i.e., for some α > G ′ ( u ) ≥ α in S ε × (0 , T ] . IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 17
Select a function G such that hypothesis H2 is satisfied; moreover, G ( u ) = G ( u )for u ≥ α and G ′ ( u ) ≥ α > u ∈ R . From (3.50), u η ( x, t ) is a solution tothe non-degenerate equation ρ∂ t u = (cid:2) G ( u ) (cid:3) in S ε × (0 , T ] . Thus we get the conclusion as in the proof of Theorem 2.3 . (cid:3) uniqueness results: proofs The proof of Theorem 2.7 makes use of the following lemma.
Lemma 4.1.
Let ε > and F ∈ C ∞ (Ω) such that F ≥ , supp F ⊂ Ω ε . Then,for any < ε < ε , there exists a unique classical solution ψ ε to the problem (4.1) ∆ ψ ε = − F in Ω ε ψ ε = 0 on A ε . Moreover, for any < ε < ε there holds: (4.2) ψ ε > in Ω ε ;(4.3) h∇ ψ ε ( x ) , ν ε ( x ) i < for all x ∈ A ε ;(4.4) Z A ε (cid:12)(cid:12) h∇ ψ ε , ν ε i (cid:12)(cid:12) dS ≤ ¯ C , for some constant ¯ C > independent of ε ; here ν ε denotes the outer unit normalvector to ∂ Ω ε .Proof. For any 0 < ε < ε , the existence and the uniqueness of the solution ψ ε to(4.1) follow immediately. Moreover, since F ≥
0, by the strong maximum principlewe get (4.2) and (4.3). Observe that, since supp F ⊂ Ω ε , then for any 0 < ε < ε we have(4.5) Z Ω ε F ( x ) dx = Z Ω ε F ( x ) dx =: ¯ C .
On the other hand, from (4.1) by integrating by parts,(4.6) Z Ω ε F ( x ) dx = − Z Ω ε ∆ ψ ε dx = − Z A ε h∇ ψ ε , ν ε i dS . From (4.5), (4.6), and (4.3) we get (4.4) . (cid:3)
Proof of Theorem 2.7.
In view of the hypotheses we made, we can apply The-orem 2.3 to infer that there exists a maximal solution ¯ u to (2.1). Let u be anysolution to (2.1), and let F ∈ C ∞ c (Ω).Without loss of generality, we suppose supp F ⊂ Ω ε , for some ε > F F ≥
0. Since both ¯ u and u solves (2.1), we apply the equality (2.2) with Ω = Ω ε ,0 < ε < ε and ψ ( x, t ) = ψ ε ( x ). We get Z T Z Ω ε [ G (¯ u ) − G ( u )] F ( x ) dx dt == − Z Ω ε [¯ u ( x, T ) − u ( x, T )] ρ ( x ) ψ ε ( x ) dx − Z T Z A ε [ G (¯ u ) − G ( u )] h∇ ψ ε , ν ε i dS dt Since F ≥ , ψ ε ≥
0, ¯ u ≥ u in Ω ε and h∇ ψ, ν ε i ≤ A ε , the previous equalitygives:(4.7) Z T Z Ω ε [ G (¯ u ) − G ( u )] F ( x ) dx dt ≤ − Z T Z A ε [ G (¯ u ) − G ( u )] h∇ ψ, ν ε i dS dt == − Z τ Z A ε [ G (¯ u ) − G ( u )] h∇ ψ, ν ε i dS dt − Z Tτ Z A ε [ G (¯ u ) − G ( u )] h∇ ψ, ν ε i dS dt Going further, by (4.4), we get(4.8) Z Tτ Z Ω ε [ G (¯ u ) − G ( u )] F ( x ) dx dt ≤ sup A ε × ( τ,T ) [ G (¯ u ) − G ( u )] Z A ε (cid:12)(cid:12) h∇ ψ, ν ε i (cid:12)(cid:12) dS dt ≤ ¯ C sup A ε × ( τ,T ) [ G (¯ u ) − G ( u )] . Furthermore(4.9) Z τ Z Ω ε [ G (¯ u ) − G ( u )] F ( x ) dx dt ≤ ¯ C τ C, where the constant C only depends on k u k L ∞ and k ¯ u k L ∞ . Since any solution to(2.1) satisfies condition (2.5) uniformly for t ∈ [ τ, T ], for each τ ∈ (0 , T ), we get(4.10) sup A ε × ( τ,T ) [ G (¯ u ) − G ( u )] → ε → . Hence, in view of (4.8), (4.9) and (4.10), if we let ε → τ → Z T Z Ω (cid:2) G (¯ u ) − G ( u ) (cid:3) F ( x ) dx dt = 0 . In view of the hypothesis H2 , and because of the arbitrariness of F , (4.11) implies¯ u = u in Ω × (0 , T ] , and the proof is completed. (cid:3) IRICHLET BOUNDARY CONDITIONS FOR NONLINEAR PARABOLIC EQUATIONS 19
As outlined in Remark 2.8, Theorem 2.7 holds true either if we consider a nondegenerate nonlinearity G satisfying hypothesis H5 or if we suppose ϕ ( x , t ) ≡ ϕ ( x ) , for all t ∈ [0 , T ] . Infact, in both cases, Theorem 2.3 and Theorem 2.5 assure the existence of themaximal solution satisfying (2.5) and (2.6) respectively. Hence, the uniquenessfollows as in the proof of Theorem 2.7.
References [1] D. Aronson, M.C. Crandall, L.A. Peletier,
Stabilization of solutions of a degenerate nonlineardiffusion problem , Nonlin. Anal. TMA (1982), 1001-1022 .[2] E. Di Benedetto, Continuity of weak solutions to a general porous medium equation,
IndianaUniv. Math. J. (1983), 83–118 .[3] E. Di Benedetto, A Boundary modulus of continuity for a class of singular parabolic equations,
J. Diff. Eq. (1986), 418–447 .[4] D. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneousmedium , J. Differential Equations (1990), 309–318.[5] D. Eidus, S. Kamin, The filtration equation in a class of functions decreasing at infinity , Proc.Amer. Math. Soc. (1994), 825–830.[6] P. M. N. Feehan, C. A. Pop,
Schauder a priori estimates and regularity of solutions toboundary-degenerate elliptic linear second-order partial differential equations , J. Diff. Eq. (2014), 895–956.[7] P. M. N. Feehan,
Maximum principles for boundary-degenerate second order linear ellipticdifferential operators , Comm. Partial Diff. Eq. (2013), 1863–1935.[8] P. M. N. Feehan, C. A. Pop, A Schauder approach to degenerate-parabolic partial differentialequations with unbounded coefficients , J. Diff. Eq. (2013), 4401–4445.[9] G. Fichera,
Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,
AttiAccad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I (1956), 1-30.[10] G. Fichera, On a unified theory of boundary value problem for elliptic-parabolic equations ofsecond order . In:
Boundary Value Problems in Differential Equations , pp. 97-120 (Universityof Wisconsin Press, 1960).[11] M. Freidlin,
Functional Integration and Partial Differential Equations (Princeton UniversityPress, 1985).[12] A. Friedman,
Partial Differential Equations of Parabolic Type , Prentice-Hall Inc., EnglewoodCliffs, N.J., 1964.[13] D. Gilbarg, N.S. Trudinger,
Elliptic Partial Differential Equations of Second Order ,[14] G. Grillo, M. Muratori, F. Punzo,
Conditions at infinity for the inhomogeneous filtrationequation , Ann. I. H. Poincar-AN (2014), 413–428 .[15] M. Guedda, D. Hilhorst, M.A. Peletier, Disappearing interfaces in nonlinear diffusion , Adv.Math. Sc. Appl. (1997), 695–710.[16] A. M. Il’in, A. S. Kalashnikov, O. A. Oleinik, Linear equations of the second order of parabolictype , Russian Math. Surveys (1962), 1–144.[17] R. Kersner, A. Tesei, Well-posedness of initial value problems for singular parabolic equations ,J. Differ. Equations 199 (2004), 47–76.[18] S. Kamin, M.A. Pozio, A. Tesei,
Admissible conditions for parabolic equations degeneratingat infinity , St. Petersburg Math. J. (2008), 239–251.[19] S. Kamin, F. Punzo, Prescribed conditions at infinity for parabolic equations , Comm. Cont.Math. (to appear), DOI: 10.1142/S0219199714500047 .[20] S. Kamin, F. Punzo,
Dirichlet conditions at infinity for parabolic and elliptic equations ,preprint 2014. [21] S. Kamin, P. Rosenau,
Propagation of thermal waves in an inhomogeneous medium , Comm.Pure Appl. Math. 34 (1981), 831–852.[22] S. Kamin, P. Rosenau,
Non-linear diffusion in a finite mass medium , Comm. Pure Appl.Math. (1982), 113–127.[23] R.Z. Khas’minskii, Diffusion processes and elliptic differential equations degenerating at theboundary of the domain , Th. Prob. Appl. (1958), 400-419.[24] O.A. Ladyzhenskaya, V.A. Solonnikov, N.A. Uraltseva, ”Linear and Quasilinear Equationsof Parabolic Type”, Nauka, Moscow (1967) (English translation: series Transl. Math. Mono-graphs, AMS, Providence, RI, 1968) .[25] K. Miller,
Barriers on Cones for Uniformly Elliptic Operators , Ann. Mat. Pura e Appl., ,1976, 93–105[26] D. D. Monticelli, K. R. Payne, Maximum principles for weak solutions of degenerate ellipticequations with a uniformly direction , J. Diff. Eq. (2009), 1993–2026 .[27] O.A. Oleinik, E.V. Radkevic,
Second Order Equations with Nonnegative Characteristic Form (Amer. Math. Soc., Plenum Press, 1973).[28] M.A. Pozio, F. Punzo, A. Tesei,
Criteria for well-posedness of degenerate elliptic and parabolicproblems , J. Math. Pures Appl. (2008), 353–386.[29] M.A. Pozio, F. Punzo, A. Tesei, Uniqueness and nonuniqueness of solutions to parabolicproblems with singular coefficients , Discr. Cont. Dyn. Sist.-A (2011), 891–916 .[30] F. Punzo, On the Cauchy problem for nonlinear parabolic equations with variable density , J.Evol. Equations (2009), 429-447.[31] F. Punzo, Uniqueness and nonuniqueness of bounded solution to singular nonlinear parabolicequations,
Nonlin. Anal. TMA (2009), 3020–3029.[32] K. Taira, Diffusion Processes and Partial Differential Equations (Academic Press, 1998).2