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On approximate self-similar solutions of a class of quasilinear heat equations with a source. (English. Russian original) Zbl 0573.35049
Math. USSR, Sb. 52, 155-180 (1985); translation from Mat. Sb., Nov. Ser. 124(166), No. 2(6), 163-188 (1984).
Non-stationary dissipative structures resulting from ”nonlinear interaction” of heat diffusion and energy release proceses and given by quasilinear parabolic equations of the form \[ (A)\quad \partial u/\partial t=\nabla (k(u)\nabla u)+Q(u),\quad k>0 \] are analyzed. It is shown that for a wide class of functions k and Q the asymptotic behaviour of solutions to the boundary-value problem for the equation (A) as \(t\to +\infty\) is connected with the invariant solutions to the first order nonlinear equation of Hamilton-Jacobi type \[ \partial u_ a/\partial t=(k(u_ a)/(u_ a+1))(\nabla u_ a)^ 2+Q(u_ a). \] Moreover it turns out that these invariant solutions are so-called approximate self- similar solutions which do not satisfy (A) but to which the solution of the appropriate problem converges asymptotically in some special norms [cf. the first and third authors, Mat. Sb., Nov. Ser. 118(160), 291-322, 435-455 (1982; Zbl 0529.35043, Zbl 0529.35044); 120(162), No.1, 3-21 (1983; Zbl 0529.35045)].
Reviewer: I.Zino

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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