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Blow-up behaviour of one-dimensional semilinear parabolic equations. (English) Zbl 0813.35007

Summary: Consider the Cauchy problem \[ u_ t - u_{xx} - F(u) = 0; \quad x \in \mathbb{R}, \quad t > 0, \qquad u(x,0) = u_ 0(x); \quad x \in \mathbb{R} \] where \(u_ 0(x)\) is continuous, nonnegative and bounded, and \(F(u) = u^ p\) with \(p>1\), or \(F(u) = e^ u\). Assume that \(u\) blows up at \(x = 0\) and \(t = T > 0\). In this paper we describe the various possible asymptotic behaviours of \(u(x,t)\) as \((x,t) \to (0,T)\). Moreover, we show that if \(u_ 0(x)\) has a single maximum at \(x=0\) and is symmetric, \(u_ 0(x) = u_ 0 (-x)\) for \(x>0\), there holds
1) If \(F(u) = u^ p\) with \(p>1\), then \[ \begin{split} \lim_{t \uparrow T} u \biggl( \xi \bigl( (T-t) | \log (T - t) | \bigr)^{1/2}, t \biggr) \\ \times (T-t)^{1/(p-1)} = (p-1)^{-(1/(p-1))} \left[ 1 + {(p-1) \xi^ 2 \over 4p} \right]^{-(1/(p-1))} \end{split} \] uniformly on compact sets \(| \xi | \leq R\) with \(R>0\),
2) If \(F(u) = e^ u\), then \[ \lim_{t \uparrow T} \Bigl( u \biggl( \xi \bigl( (T-t) | \log (T-t) | \bigr)^{1/2}, t \biggr) + \log (T - t) \Bigr) = - \log \left[ 1 + {\xi^ 2 \over 4} \right] \] uniformly on compact sets \(| \xi | \leq R\) with \(R>0\).

MSC:

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

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