## 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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