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)


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