## Solution of a nonlinear heat equation with arbitrarily given blow-up points.(English)Zbl 0785.35012

Summary: We consider the equation $u(0,x)=\varphi(x),\;u_ t=u_{xx}+| u|^{p-1}u\text{ on }[0,T)\times I,\;u=0\text{ on }[0,T)\times\partial I, \tag{1}$ where $$I\subset\mathbb{R}$$, $$u$$ is scalar- valued and $$p>1$$. It has been proven that if $$u(t)$$ blows up at time $$T$$, the blow-up points are finite in number and located in $$\overset\circ I$$.
Our aim is to prove that this result is optimal. That is, for any given points $$x_ 1,\ldots,x_ k$$ in $$\overset\circ I$$, there is a solution $$u$$ such that its blow-up points are exactly $$x_ 1,\dots,x_ k$$.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B35 Stability in context of PDEs

### Keywords:

blow-up points; semilinear heat equation
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### References:

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