Abstract

In this paper, we are concerned with the following initial-boundary value problem: (P) \left\{% \begin{array}{ll} \hbox{$u_t(x,t)- \Delta u(x,t)- G(x)|u|^{p-1}u  =0, \quad x\in  \Omega, t\in(0,T)$,} \hbox{$u(x,t)=0 \quad x \in \partial \Omega,  t\in(0,T)$,} \hbox{$u(x,0)=u_{0}(x), \quad x \in \Omega,$} \\ \end{array}% \right. where $ p \geq p_s :=\dfrac{ d + 2}{d-2} $, $ u_0 \in L^\infty(D_\mu) $, and $ G(r) \in C^1([0, \mu]), $  $0 <  \underline{C} \leq G(r) \leq \overline{C}  < \infty,$ $G^{'}(r) \leq 0 $. We study the initial value problem and boundary conditions for a nonlinear heat equation incorporating a potential term. Particularly, we focus on the asymptotic behavior of solutions during blow-ups. We extend existing results on this phenomenon, specifically in the case where the potential term is constant, based on the works of Matano-Merle (Matano  \& Merle, 2004). We  show that when $p_s \leq p < p^* $, the radial solutions of this problem always exhibit Type I blow-up. This result generalizes previous results for the case where $G \equiv 1 $, and its achievement is non-trivial due to the presence of the potential term $ G $. We use the contraction mapping principle to  show the existence of singular stationary solutions to an associated elliptic equation with a potential. Furthermore, our analysis of the properties of the zeros of the solutions lead to the nonexistence of type II singularity for the problem. We also delve into the study of critical solutions for a class of nonlinear  parabolic equations in a bounded domain, focusing on the construction of appropriate approximate solutions.

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