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Boundedness up to blow-up of the difference between two solutions to a semilinear heat equation. (English) Zbl 0954.35085

We consider \(u_1\) and \(u_2\), two blow-up solutions of \(\partial_t u=\Delta u+|u|^{p-1}u\), \((x,t)\in {\mathbb{R}}^N \times [0,T)\), where \(p>1,\) \((N-2)p<N+2\) and either \(u(0)\geq 0\) or \((3N-4)p<3N+8\). We assume that \(u_1\) and \(u_2\) blow-up at the same time \(T\), at the same unique point \(a\in {\mathbb{R}}^N\) and that they have the same (generic) profile. We then obtain a sharp estimate on \(|u_1-u_2|\) for all \((x,t)\in {\mathbb{R}}^N\times [0,T).\) In particular, we show that, up to a scaling, this difference is uniformly bounded and goes to zero as \((x,t)\to (a,T)\), provided \(N=1\) and \(p\geq 3\). As an application of our result, we show the stability of the considered profile in \(N\) dimensions.

MSC:

35K57 Reaction-diffusion equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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