## Advances in Differential Equations

- Adv. Differential Equations
- Volume 15, Number 1/2 (2010), 73-98.

### Self-similarity and uniqueness of solutions for semilinear reaction-diffusion systems

Lucas C.F. Ferreira and Eder Mateus

#### Abstract

We study the well posedness of the initial-value problem for a coupled semilinear reaction-diffusion system in Marcinkiewicz spaces $L^{(p_{1}, \infty)}(\Omega)\times L^{(p_{2}, \infty)}(\Omega)$. The exponents $p_{1},p_{2}$ of the initial-value space are chosen to allow the existence of self-similar solutions (when $\Omega=\mathbb{R}^{n}$). As a nontrivial consequence of our coupling-term estimates, we prove the uniqueness of solutions in the scaling invariant class $C([0,\infty);L^{p_{1}}(\Omega)\times L^{p_{2}}(\Omega))$ regardless of their size and sign. We also analyze the asymptotic stability of the solutions, show the existence of a basin of attraction for each self-similar solution and that solutions in $L^{p_{1} }\times L^{p_{2}}$ present a simple long-time behavior.

#### Article information

**Source**

Adv. Differential Equations, Volume 15, Number 1/2 (2010), 73-98.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355854764

**Mathematical Reviews number (MathSciNet)**

MR2588390

**Zentralblatt MATH identifier**

1198.35047

**Subjects**

Primary: 35K55: Nonlinear parabolic equations 35A05 35B 35B40: Asymptotic behavior of solutions

#### Citation

Ferreira, Lucas C.F.; Mateus, Eder. Self-similarity and uniqueness of solutions for semilinear reaction-diffusion systems. Adv. Differential Equations 15 (2010), no. 1/2, 73--98. https://projecteuclid.org/euclid.ade/1355854764