## 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.

### MSC:

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