Large Time Behavior Of Solutions To Semilinear Parabolic Equations With Gradient

In1966, Fujita [4] proved the following results. The Cauchy problemof the heat equationsdoes not have any nontrivial global nonnegative solution if1<p <pc=, whereas there exist both non-global nonnegative and nontrivialglobal solutions if p> pc. We call pcthe critical Fujita exponent andsuch a result a blowing-up theorem of Fujita type. Moreover, for p=pc,Hayakawa [5] and Kobayashi [6] show the results for n=1,2and forall n≥1that the problem possesses no nontrivial global nonnegativesolution. There have been a number of extensions of Fujita’s results inseveral directions, after the pioneering work [4], including similar resultsfor systems in various of geometries (whole spaces, cones and exterior do-mains) with nonlinear sources or nonhomogeneous boundary conditions.and numerous of quasilinear parabolic equations. We investigate the largetime behavior of solutions to the Cauchy problem of a class of semilinearparabolic equations with gradient in this paper. The blowing-up theoremof Fujita type is established and the critical Fujita exponent is formulat-ed by the behavior of the coefcient of the gradient term at∞and thespacial dimension. What make people surprise is that due to the efect ofthe gradient term, the critical Fujita exponent even could be1or infinite. The critical case is also considered. An interesting phenomenon is thatfor each nontrivial solution the critical Fujita exponent can belong to notonly the blowing-up case but also the global existence case. We determinethe critical Fujita exponent to the Cauchy problem of (1) in this paperexplicitly, when b is radial symmetric. Now, we consider the problemwhere0≤u0∈L∞(Rn), b∈C1([0,+∞)) satisfiesinf{s2b(s): s>0}> n if n <κ≤+∞.(3)It is proved that to the problem (1),(2) the critical Fujita exponent canbe formulated asFor the problem (1),(2), from (4) we can know that the gradient term canafect the large time behavior of solutions essentially. The spacial dimen-sion together with the behavior of the coefcient of the gradient term at∞, determines precisely the critical Fujita exponent to the problem (1),(2). Furthermore, what makes people surprised is that the critical Fujitaexponent even could be1or infinite by (4). So far, this is the first time toget pc=1or pc=+∞for Cauchy problems of nonlinear parabolic equa-tions with reactions of power function type (see [2,7]). It is shown thatthe critical Fujita exponent to the problem (1),(2) is determined explic-itly by the behavior of the coefcient of the gradient term at∞and the spacial dimension in this paper. Moreover, pcis given by (4). We deter-mine the interactions among the difusion, the gradient and the reactionby a precise energy integral estimate instead of pointwise comparisons toprove the blowing-up of solutions. The key is to choose a suitable weightfor the energy integral. For the global existence of nontrivial solutions,we try to construct a nontrivial global supersolution. but we find that(1) does not possess a self-similar construct, so we have to seek a compli-cated supersolution and do some precise calculations. By the way,(3) isused only for constructing a nontrivial global supersolution and when oneconstructs such a supersolution it seems necessary. Next, we investigatethe critical case p=pc. It is clearly that for the case p=pc=1, i.e.κ=+∞, each solution to the problem (1),(2) exists globally. It is aninteresting phenomenon that pccan belong to the global existence casefor each nontrivial solution for Cauchy problems of nonlinear parabolicequations with reactions of power function type (see [2,7]). Noting from(4) that if and only if n <κ <+∞,1<pc<+∞. So, for p=pc, onlythe case n <κ <+∞needs to be considered, which is shown to be theblowing-up case for each nontrivial solution. Generally speaking, for theblowing-up of solutions, when p <pc, the efect of the difusion and thegradient has a lower grade than the efect of the reaction when p=pc,while these two kinds of efect are on the same grade. Therefore, for theblowing-up of solutions, handling the case p=pcis more difcult andcomplicated than the case p <pc. Besides, to determine the gradientfor the blowing-up of solutions and the precise grade of the efect of thedifusion, it is assumed additionally that1sds <+∞By estimating a sequence of energy integrals similar to [10,12,14], weshow the blowing-up of solutions to the problem (1),(2) for the criticalcase p=pcwith n <κ <+∞.The following is how this paper is organized. In§2we establishedthe blowing-up theorem of Fujita type for the problem (1),(2). That isto say, the problem (1),(2) does not have any nontrivial global solutionif1<p <pc, whereas there exist both nontivial global and non-global solutions if p> pc. Subsequently, in§3we treat the critical case p=pcwith-n <κ <+∞, which is shown to belong to the blowing-up case foreach nontrivial solution.