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

Carpio, Ana

C. R. Acad. Sci., Paris, Sér. I 319, No. 1, 51-56 (1994).

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