## Blow-up profiles in one-dimensional, semilinear parabolic problems.(English)Zbl 0772.35027

Summary: We consider the Cauchy problem (1) $$u_ t=u_{xx}+u^ p$$, $$x\in\mathbb{R}$$, $$t>0$$; (2) $$u(x,0)=u_ 0(x)$$, $$x\in\mathbb{R}$$ where $$p>1$$ and $$u_ 0(x)$$ is continuous, nonnegative and bounded. Let $$u(x,t)$$ be the solution of (1), (2), and assume that $$u$$ blows up at $$t=T<+\infty$$, and $$u(x,t)\not\equiv(p-1)^{-1/p-1}(T-t)^{-1/p-1}$$. We then show that the blow-up set is discrete. Moreover, if $$x=0$$ is a blow-up point, one of the two following possibilities occurs: Either $$\lim_{x\to 0}(| x|^ 2/|\log | x| |)^{1/p-1}u(x,T)=(8p/(p-1)^ 2)^{1/p-1}$$ or there exist $$C>0$$ and an even number $$m$$, $$m\geq 4$$ such that $$\lim(| x|^{m/p-1}u(x,T))=C$$.

### MSC:

 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35K57 Reaction-diffusion equations

Cauchy problem
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