## Flat blow-up in one-dimensional semilinear heat equations.(English)Zbl 0767.35036

Consider the Cauchy problem $u_ t=u_{xx}+u^ p, \quad x\in\mathbb{R}, \quad t>0; \qquad u(x,0)=u_ 0(x), \quad x\in\mathbb{R},$ where $$p>1$$ and $$u_ 0$$ is continuous, nonnegative and bounded. Assume that $$u(x,t)$$ blows up at $$x=0$$, $$t=T$$.
The authors show that there exist initial values $$u_ 0$$ for which the corresponding solution is such that two maxima collapse at $$x=0$$, $$t=T$$. The asymptotic behaviour is different and flatter than that corresponding to solutions spreading from data $$u_ 0$$ having a single maximum.
Reviewer: P.Bolley (Nantes)

### MSC:

 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations 35K05 Heat equation 35K15 Initial value problems for second-order parabolic equations

### Keywords:

semi-linear heat equation; Cauchy problem