## Uniqueness and asymptotic behaviour for scalar convection-diffusion equations. (Unicité et comportement asymptotique pour des équations de convection-diffusion scalaires.)(French. Abridged English version)Zbl 0806.35068

Summary: We prove the uniqueness of the fundamental entropy solutions $$u(x,y,t)$$ of the equation: $u_ t - \Delta_ x u + \partial_ y \bigl( | u |^{q - 1} u \bigr) = 0 \quad \text{in} \quad \mathbb{R}^{n-1} \times \mathbb{R} \times \mathbb{R}^ + \tag{R}$ when $$1 < q < 1 + (2/(n-1))$$ if $$n>2$$ and $$1 < q \leq 2$$ if $$n = 1,2$$. As a consequence, we prove that the large time behaviour of solutions to the equation $u_ t - \Delta_ x u - \partial^ 2_{yy}u + \partial_ y \bigl( | u |^{q-1} u \bigr) = 0, \quad \text{in } \quad \mathbb{R}^{n-1} \times \mathbb{R} \times \mathbb{R}^ +$ with initial data $$u_ 0 \in L^ 1(\mathbb{R}^ n)$$ is given by the fundamental solutions of $$(R)$$ with mass $$\int u_ 0$$ when $$1 < q < 1 + (1/n)$$. This completes a result by Escobedo, Vazquez and Zuazua for positive solutions.

### MSC:

 35K55 Nonlinear parabolic equations 35A08 Fundamental solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

fundamental entropy solutions