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On the method of stationary states for quasilinear parabolic equations. (English. Russian original) Zbl 0701.35010
Math. USSR, Sb. 67, No. 2, 449-471 (1990); translation from Mat. Sb. 180, No. 8, 995-1016 (1989).
The authors study the Cauchy problem of the quasilinear parabolic equations with source term, i.e. $u_ t=\nabla \cdot (1+| \nabla u|^ 2)^{\sigma /2} \nabla u)+u^{\beta},\quad t>0,\quad x\in R,$ $u(x,0)=u_ 0(x)\geq 0,\quad x\in R^ N;\quad \sup u_ 0<+\infty,\quad u_ 0\in C(R).$ They show that when $$\sigma >0$$, $$\beta >1$$ and the solution u(t,x) of the above Cauchy problem satisfies condition A, then $\quad \limsup_{t\to T^-_ 0}\sup_{| x| =R}u(t,x)=+\infty,\quad \forall \beta \in (1,\sigma +1),\quad R>0,$ $(ii)\quad \limsup_{t\to T^-_ 0}\sup_{| x| =R}u(t,x)=+\infty,\quad \forall R\in (0,[(\sigma +2/\sigma +1)^{\sigma +1}N]^{(1/\sigma +2)}),\quad \beta =\sigma +1,$ (iii) when $$\beta >\sigma +1$$, for any arbitrarily small $$R>0$$, there exists $$t_ k\equiv (0,T_ 0)$$, such that $\sup_{| x| =0}u(t_ k,t)>A_ 0R^{-(\sigma +2)/(\beta -(\sigma +1))},$ where $$A_ 0$$ is a constant depending on $$\beta$$, $$\sigma$$ and N.
The method of stationary states is used. Similar problems for blow-up of solutions for the initial-boundary problem and parabolic systems of quasilinear equations are discussed.
Reviewer: J.Wang

##### MSC:
 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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