## 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

### Keywords:

stability of profiles; singular points
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