Asymptotic behaviour and source-type solutions for a diffusion-convection equation
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 \(q>1\).
scientific article; zbMATH DE number 474500
scientific article