## Advances in Differential Equations

- Adv. Differential Equations
- Volume 15, Number 3/4 (2010), 283-314.

### Asymptotic behavior of solutions of a semilinear heat equation with localized reaction

#### Abstract

We consider non-negative solutions to the Dirichlet problem of a semilinear heat equation with localized reaction in $\Omega$: $u_t = \Delta u +f(u(x_0(t),t))$, where $\Omega$ is a smooth bounded domain, $x_0(t)$ is a locally Hölder continuous function from $[0,\infty)$ into $\Omega$ and $f$ satisfies $f(0)=f'(0)=0$ and some blow-up condition. We show that, if $x_0(t)$ remains in some compact subset of $\Omega$ as $t\to\infty$, then all global solutions are bounded in $\Omega\times(0,\infty)$ and, if $x_0(t)$ approaches the boundary of $\Omega$ as $t\to \infty$, then some unbounded global solution (infinite time blow-up solution) exists. These results are parts of our main results on the classification of all solutions.

#### Article information

**Source**

Adv. Differential Equations, Volume 15, Number 3/4 (2010), 283-314.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355854751

**Mathematical Reviews number (MathSciNet)**

MR2588771

**Zentralblatt MATH identifier**

1198.35128

**Subjects**

Primary: 35B35: Stability 35B40: Asymptotic behavior of solutions 35B45: A priori estimates 35K20: Initial-boundary value problems for second-order parabolic equations 35K55: Nonlinear parabolic equations 35K57: Reaction-diffusion equations

#### Citation

Suzuki, Ryuichi. Asymptotic behavior of solutions of a semilinear heat equation with localized reaction. Adv. Differential Equations 15 (2010), no. 3/4, 283--314. https://projecteuclid.org/euclid.ade/1355854751