Advances in Differential Equations

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

Michael Winkler

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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 $t\to\infty$.

Article information

Adv. Differential Equations Volume 13, Number 1-2 (2008), 27-54.

First available in Project Euclid: 18 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K57: Reaction-diffusion equations
Secondary: 35B40: Asymptotic behavior of solutions 35K65: Degenerate parabolic equations


Winkler, Michael. Conservation of boundary decay and nonconvergent bounded gradients in degenerate diffusion problems. Adv. Differential Equations 13 (2008), no. 1-2, 27--54.

Export citation