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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

Keywords:

Cauchy problem
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