Conservation of boundary decay and nonconvergent bounded gradients in degenerate diffusion problems

Abstract

This paper investigates the boundary behavior of nonnegative classical solutions to the Dirichlet problem for $$ u_t=u^p\Delta u + g(u) \qquad \mbox{in } \Omega \times (0,T), \qquad p>1, $$ and draws some consequences for the large time behavior of solutions. Here, $\Omega \subset \mathbb R^n$ is a smooth bounded domain and $g:\ [0,\infty) \to \mathbb R$ is locally Lipschitz continuous with $g(0)=0$. The first goal is to study for which $\alpha>0$ the implication \begin{align} & & u_0(x) \le c_1 ({{\rm dist} \, } (x,{\partial\Omega}))^\alpha \qquad (c_1>0) \notag \\ & \Rightarrow & u(x,t) \le C(T') ({{\rm dist} \, }(x,{\partial\Omega}))^\alpha \quad \mbox{in } \Omega\times (0,T') \ \mbox{for any } T' < T, \qquad \mbox{(I)} \tag*{(I)} \end{align} is valid, and it turns out that this holds whenever either $p \ge 2$, or $p < 2$ and $\alpha \ge \frac{1}{p-1}$. For $p \in (1,2]$ and $g\equiv 0$, this complements a previously known result, according to which the lower estimate $u_0(x)\ge c_0(\text{dist}\, (x,{\partial\Omega}))^\alpha$ with some $\alpha <\frac{1}{p-1}$ and $c_0>0$ implies the existence of $T>0$ and $C>0$ such that $u(x,t) \ge C \text{dist}\, (x,{\partial\Omega})$ for all $x \in \Omega$ and $t \ge T$. Using (I), we moreover show that whenever $p>1$, there exist some values of $q \ge 1$ such that the particular equation, $u_t=u^p u_{xx}+u^q$, possesses positive classical solutions which are nondecreasing w.r.~to $t$ and remain uniformly bounded in $C^1(\bar\Omega)$ for all times, but do not converge in $C^1(\bar\Omega)$ as $t\to\infty$.