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Higher dimensional blow up for semilinear parabolic equations. (English) Zbl 0813.35009

Summary: We consider the Cauchy problem \[ u_ t = \Delta u + u^ p\quad \text{if}\quad x \in \mathbb{R}^ N, \;t > 0, \quad u(x,0) =u_ 0(x) \quad \text{if} \quad x \in \mathbb{R}^ N, \] where \(N \geq 1\), \(p>1\) and \(u_ 0(x)\) is a continuous, nonnegative and bounded function. It is well known that the solution \(u(x,t)\) may blow up in a finite time \(T < + \infty\). We shall be concerned here with the asymptotic behaviour of \(u(x,t)\) as blow up is approached. In particular the final profile of \(u(x,T)\) near blow up points is studied, and the fact that the blow up set has zero Lebesgue measure is proved.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K57 Reaction-diffusion equations
35K15 Initial value problems for second-order parabolic equations
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References:

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