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The space structure near a blow-up point for semilinear heat equations: A formal approach. (English) Zbl 0747.35014
Comput. Math. Math. Phys. 31, No. 3, 46-55 (1991); and Zh. Vychisl. Mat. Mat. Fiz. 31, No. 3, 399-411 (1991).
The authors study the blow-up of a positive solution \(u\) to the parabolic equation \(u_ t-u_{xx}=f(u)\), \(-\infty<x<+\infty\), \(t>0\), where either \(f(u)=\exp u\) or \(f(u)=u^ p\) \((p>1)\). The solution \(u\) is assumed to have a single-point blow-up at \(x=0\) and \(t=T\). By means of formal expansions, a description of \(u\) near the blow-up point is proposed.

35K55 Nonlinear parabolic equations
35K05 Heat equation
74A40 Random materials and composite materials
35B40 Asymptotic behavior of solutions to PDEs