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Asymptotic behaviour and source-type solutions for a diffusion-convection equation. (English) Zbl 0807.35059

The paper deals with the equation \(u_ t = u_{xx} - | u |^{q - 1} u_ x\), \(t > 0\), \(x \in \mathbb{R}\). The asymptotic behaviour of the solutions of this equation is studied in the subcritical exponent range, that is, in the case \(1<q<2\). It is proved that as \(t \to \infty\) the solutions with integrable data look like the entropy solutions of the equation \(u_ t + (| u |^{q - 1} u/q)_ x = 0\) with initial data \(M \delta (x)\). Also the authors investigate the question of existence of fundamental solutions of the equation under consideration and they construct fundamental solutions for all exponents \(q>1\).
Reviewer: E.Minchev (Sofia)

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients
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