## Differential and Integral Equations

- Differential Integral Equations
- Volume 19, Number 12 (2006), 1349-1370.

### Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations

Lucas Catão de Freitas Ferreira and Elder Jesús Villamizar-Roa

#### Abstract

We analyze the well-posedness of the initial-value problem for the semilinear equation in Marcinkiewicz spaces $L^{(p,\infty)}$. Mild solutions are obtained in spaces with the right homogeneity to allow the existence of self-similar solutions. As a consequence of our results we prove that the class $C([0,T);L^{p}(\Omega)),\ 0 < T\leq\infty, \ p={\frac{n(\rho-1)}{2\gamma }},$ $\Omega=R^{n},$ has uniqueness of solutions (including large solutions) obtained in [19], [20] and [8]. The asymptotic stability of solutions is obtained, and as a consequence, a criterion for self-similarity persistence at large times is obtained.

#### Article information

**Source**

Differential Integral Equations, Volume 19, Number 12 (2006), 1349-1370.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356050293

**Mathematical Reviews number (MathSciNet)**

MR2279332

**Zentralblatt MATH identifier**

1212.35205

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35B40: Asymptotic behavior of solutions 35C05: Solutions in closed form 35K15: Initial value problems for second-order parabolic equations

#### Citation

Ferreira, Lucas Catão de Freitas; Villamizar-Roa, Elder Jesús. Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations. Differential Integral Equations 19 (2006), no. 12, 1349--1370. https://projecteuclid.org/euclid.die/1356050293