Liouvilie Type Results for a Class of Quasilinear Parabolic Problem

The mathematical theory of Blow-up began in the sixties of the latest century, with the works of Fujita and Friedman [1-3]. The two main models considered in those works were the semilinear heat equation with f(u)=up and f(u)=eu. From those days, an increasing interest on blow-up problems has attracted a great number of researchers [4-6]. Equations of the type (1.1) arise especially in population dynamics and ecological models, where the nonnegative quantity u typically stands for the concentration of a species [7-9]. Stationary problems associated to (1.1)1 arises in many nonlinear phenomena, for instance, in the theory of quasi-regular and quasi-conformal mappings [10-12] and in mathematical modeling if nonNewtonian fluids, see [13-15] for a discussion of the physical background. Our study in the present work was partly motivated first by Belhaj et al. [16], in which the authors treated the problem (1.1) in the stationary case and the papers [17,18] concerning the long time asymptotic behavior of a parabolic logistic equation with a degenerate spatial-temporal coefficient, of the form


Introduction
Let Ω a bounded domain in R N , (N ≥ 3) with boundary ∂Ω. In this work we consider the boundary blow-up parabolic problem: The nonlinearity f is assumed to fulfil either uniformly for any t ∈ (0,T). We denote that u=∞ on Ω ×{0} means that The mathematical theory of Blow-up began in the sixties of the latest century, with the works of Fujita and Friedman [1][2][3]. The two main models considered in those works were the semilinear heat equation with f(u)=u p and f(u)=e u. From those days, an increasing interest on blow-up problems has attracted a great number of researchers [4][5][6]. Equations of the type (1.1) arise especially in population dynamics and ecological models, where the nonnegative quantity u typically stands for the concentration of a species [7][8][9]. Stationary problems associated to (1.1)1 arises in many nonlinear phenomena, for instance, in the theory of quasi-regular and quasi-conformal mappings [10][11][12] and in mathematical modeling if non-Newtonian fluids, see [13][14][15] for a discussion of the physical background. Our study in the present work was partly motivated first by Belhaj et al. [16], in which the authors treated the problem (1.1) in the stationary case and the papers [17,18] concerning the long time asymptotic behavior of a parabolic logistic equation with a degenerate spatial-temporal coefficient, of the form In the author studies [6] the existence and the existence of blow-up parabolic problem where satisfies the assumption in the following form

Abstract
In this paper, we investigate the parabolic logistic equation with blow-up initial and boundary values on a smooth bounded domain Under suitable assumptions on a(x, t) and f, we show that such solution stays bounded in any compact subset of Ω as t increase to T. Other asymptotic estimates will given in this work.

Theorem 1.4
Under condition (1.2),(F1) and (KO), problem (1.1) has a maximal solution u and the minimal positive solution u in the sense that any positive solution u of (1.1) satisfies u u u ≤ ≤ furthermore, for any t o ∈(0,T) there exist positive constants c 1 and c 2 depending on t o , such that, for (x,t)∈ Ω×[0,T] ( ) ( ) ( ) Our result of uniqueness of solution of (1.1) is the following Theorem 1.5 Let f be a concave function on [0,∞). Then under assumptions of Theorem 1.4, the problem (1.1) has a unique positive solution.

Preliminaries
We start this section by some preliminary results. For the reader's convenience, we recall the defination of supersolution and the subsolution of (1.1). To this end, letΩ denotes a bounded domain of R N .
In the sequel, we consider a real number R>0 and x o ∈R N : The first preliminary result we need is a standard comparison principle which will be used frequently in this paper.

Proposition 2.2
Let u be a subsolution of (1.1) and v be a supersolution of (1.1) such that Suppose that f(v(x,t)) ≥ f(u(x,t)) on the set u(x,t) ≤ v(x,t), then u(x, t) ≤ v(x, t); for all x ∈Ω and t (0, T).
Using the fact that u is a subsolution and v is a supersolution of (1.1) and testing in (1.1) by (φ=u-v) + (.; t); we obtain: In this note, we will also study the qualitative properties of the global equation Throughout this study, we will make reference to the assumptions: Assume that f satisfies condition (F 2 ) and there exists c 2 >0 and γ<2 such that Then there exist a positive constant C such that any solution u of (1.1) satisfies where q>1 and the real b is a positif.
Under the hypothesis of Theorem 1.1, by extending d(x, ∂Ω) to the infinity, we obtain our second result.

Theorem 1.2
Assume that f satisfies condition (F 2 ) and there exists c 2 >0 and <2 such that Then there exist a positive constant C such that any solution u of (1.10) satisfies where q>1 and the real b is a positif.
Note that the proof Theorem 1.1 and Theorem 1.2 and other preliminary result are the aim of the second section. The third paragraph is devoted to the study of explosive solution (1.1). If we set It's well known that Ξ is a basis of open sets in X. For V∈ Ξ we denote by ∂ h V the heat boundary of V i.e. the following set The result interesting existence of explosive solutions is the following

Theorem 1.3
Suppose that there exists m>1, satisfying Combining eqns. (2.3) and (2.4), we get On the other hand, we have, Hence, we get, Then the function in non-increasing on the set (0,T) and we get Since u and v are continuous, therefore φ + =(u-v) + (., t)=0 and u(x, t) ≤ v(x, t), for all x ∈Ω and t ∈ (0; T): as required.
In the sequel, we introduce the function h on (0,1) which is very useful for the proof of Theorem 1.1.
Note the function h satisfies on ( ) ( ) 2 is well known [3] for the first half of this Proposition. It holds on a more general setting allowing f to depend on x; t and u, and for weak super and sub-solutions of (1.1) which are unbounded but satisfy certain growth conditions near x=1. We refer the interested readers [19] in which Lemma 2.3 is proved as a special case by using a maximum principle in a study [1] (Chapter2, Theorem 9) and the monotone iteration method in Theorem 3.1 of a study [19]. To obtain information about the solution, we will need the following result, which plays an important role in the proof of our main result. To this end, let denote a bounded domain of R n and we consider a real R≥1 and x o ∈ R n .

Lemma 2.4
There exists λ>0 and a nondecreasing function h on (0,1) such that the function v λ defined by, Let Satisfies where is a positive constant and h as in (2.7). We have for some λ>1 [20][21][22][23][24][25]. So it's an easy task to show that v λ fulfil the first part of the Lemma. By mean of a straightforward calculation we verify that (2.11) is equivalent to (2.12) Using assumption on the function h, precisely (2.8), eqn. (2.12) holds true and so (2.11) if

Proposition 2.5
Assume that f satisfes condition (F2) and there exist c 2 >0 andγ<2 such that Then there exist a positive constant C such that any solution u of (1.1) satisfies Proof: Let u a continuous solution of (1.1) and choose η> 0 such that u(x)>y 0 whenever d(x,∂Ω) ≤ wherever y o is defined as in (F1). Set given in (2.15), then the function v λ0 is a sursolution of (2.18) on B(x 0 .R) satisfying v λ0 (x 0 .0)= and for all z ∈∂Ω. Since the function r → r m is decreasing on [0,∞); and v λo ≥ 0 on the set{u ≥v λo }, it follows from Proposition 2.2 that u(x 0 ; t) ≥v λo (x 0 ; t) and we get Since R=d(x,∂Ω)/2 then we get, can write the following identity

Proof of Theorem 1.3
In what follows, we consider X=R N and L t ∂ = ∆ − ∂ , the heat operator on X. For every x o ∈ R N , r>0 and a, b ∈ R with a<b we denote by It's well known that Ξ is a basis of open sets in X. For V ∈ Ξ we denote by ∂ h V the heat boundary of V i.e. the following set whenever V=V (x 0 , r, a, b). By a study [2], we have the following Lemma.

Lemma 3.1
For every V∈ Ξ there exists a positive function u ∈ V with the following properties:

1) u and all derivatives
for i ∈{1,2,…N} are uniformly holder continuous in V.

Remarks 3.2:
The function u of the previous Lemma satisfies the following properties [26,27]: There exists a constant M which is an upper bound to u (0<u<M); .
We obtain the statement if we can choose c and β such that Using remark (40), we get And using the fact that w m ≤a(x,t)f(w), we get The property 3 in Remark (3.2) yields the statement about the limit of w at the heat boundary.
Proof of Theorem 1.3 Completed By a study [1], there exists for every n ∈ N a unique function u n on V such that ( ) ( ) Let v be a function on V with the properties of the Proposition 3.3. By the maximum principle, we have u n ≤ v for every n and we get u=supu n satisfies on V the required equation. Let v∈C 1 (V) satisfying with infinite limit at the heat boundary, by minimum principle, we have u n ≤ v for every n ∈ N. By passing to the limit as n→∞, we get u ≤ v.
Let (x 0 , t 0 ) ∈ V and since V is convex, it's a star domain at (x 0 , t 0 ) and (x 0 +r(x − x 0 ); t 0 +r2(t − t 0 )) is in V for every (x, t) 2 V and r ∈ [0, 1]. Let us consider, We can verify that By passing to the limit as r ↑ 1, we get v(x, t) ≤u(x, t).
Thus u=v, which is the required uniqueness.

Proof of Theorem 1.4
The proof of Theorem 1.4 is based on various comparison arguments [28][29][30][31]. We begin this section by some intermediary results which play an important role to the proof of the existence of minimal and maximal positive solution of the problem (1.1).

Lemma 4.1
The unique solution z * of the problem Proof: It suffices to apply Theorem 6.15 in a study [8].
The second intermediary result is the following, note that it's proof is immediately.

Lemma 4.2
The auxiliary problem has a unique solution v * given by the following formula For n ∈N * and ε>0, we consider the problem It's clear that (4.8) has a unique positive solution u n . Using the classical comparison principle, we get u n <u n +1. That is the sequence u n is strictly increasing on the set [ ] 0,T Ω × − ∈ . Now, we try to find a supersolution of (4.8) which is independent of the integer n.

Proof:
We begin by the case γ∈(-2,0). Using eqns. (4.5) and (4.7), we can find a constant M>1 sufficiently large such that U*:=M(v* + z*) satisfies, Using assumption (F1) on the function f, (4.6) and (1.2), we obtain Therefore U* is a super-solution of (4.8) and U*is independent on n. Now we prove the existence of super-solution for the case γ≥0 Set U 0 =M o (v*+z*) and Let n be an integer number. Then U* a super-solution and u n a solution of (4.8), we have U*(x; t)>u n (X; t) on ∂Ω× (0; T-ε) and on Ω for all small ϵ>0. Using the comparison principle, we obtain It should be noticed that, for any compact K of and for ϵ << 1, the function U* is bounded on the set K × (ϵ, T -ε). By standard regularity arguments, u n (x; t) → u(x; t) as n →∞ uniformly on any compact subset of Ω× (0; T), where u satisfies (1.1). Thus u , is a solution to (1.1) [32,33]. Note that u is the minimal positive solution of (1.1). Indeed, let u a positive solution of (1.1) and apply the comparision principle we ge: By passing to the limit as n→∞, we obtain where ϵ is a positive small real and we consider the following problem let us denote u ϵ the minimal positive solution of (4.10). If we apply the parabolic comparison principle, we get Moreover, we can extract a decreasing sequence ϵ n →0 satisfying n u u ∈ → as ∈ n →0 and u is a solution of (1.1). Indeed, for any positive solution u of (1.1) and using the parabolic comparison principle we obtain n u u ∈ ≥ in ( ) , n n T Ω × ∈ By passing to the limit as n→∞, we get u u ≥ , which proves that u is the maximal solution of (1.1). Note that the existence and uniqueness of z * δ was treated in a study [8] precisely Theorem 6.15.

Second step: Proof of the inequality (17)
To reach this aim, we pursue the same method used in the proof of (1.17) and we need the following two Lemma [34,35].

Lemma 4.7
The auxiliary problem has a unique solution z n , which converges uniformly on any compact set of Ω to the solution z*of (4.1).

Lemma 4.8
The auxiliary problem Remarks 4.9: 1) Remark that v * μ → v * as μ→0 where v * is the solution of (4.3). For the proof, we refer to Theorem 6.15 in a study [8].
2) Due to the fact that the fact that the function α 2 (defined in (1.2)) is continuous and positive on [0, T], we may suppose that there exists a constant K ∈ >0 such that Hence V is supersolution of Using the comparison principle, we get u ≤ V, which is contradiction with (5.1) then u u = and consequently the result of theorem holds.