## 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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### References:

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