zbMATH — the first resource for mathematics

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

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
Full Text: DOI