Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data (original) (raw)

A note on uniqueness of entropy solutions to degenerate parabolic equations in

Nodea-nonlinear Differential Equations and Applications, 2010

We study the Cauchy problem in \({\mathbb{R}^N}\) for the parabolic equation u_t+{\rm div}\,F(u)=\Delta\varphi(u),$$ which can degenerate into a hyperbolic equation for some intervals of values of u. In the context of conservation laws (the case φ ≡ 0), it is known that an entropy solution can be non-unique when F′ has singularities. We show the uniqueness of an entropy solution to the general parabolic problem for all L ∞ initial datum, under the isotropic condition on the flux F known for conservation laws. The only assumption on the diffusion term is that φ is a non-decreasing continuous function.

Diffuse measures and nonlinear parabolic equations

Journal of Evolution Equations, 2011

Given a parabolic cylinder Q = (0, T ) × Ω, where Ω ⊂ R N is a bounded domain, we prove new properties of solutions of ut − ∆pu = µ in Q with Dirichlet boundary conditions, where µ is a finite Radon measure in Q.

A note on uniqueness of entropy solutions to degenerate parabolic equations in RN

2009

We study the Cauchy problem in R N for the parabolic equation ut + div F (u) = ∆ϕ(u), which can degenerate into a hyperbolic equation for some intervals of values of u. In the context of conservation laws (the case ϕ ≡ 0), it is known that an entropy solution can be non-unique when F has singularities. We show the uniqueness of an entropy solution to the general parabolic problem for all L ∞ initial datum, under the isotropic condition on the flux F known for conservation laws. The only assumption on the diffusion term is that ϕ is a non-decreasing continuous function.

A note on uniqueness of entropy solutions to degenerate parabolic equations in [FORMULA]

Nodea Nonlinear Differential Equations and Applications, 2010

We study the Cauchy problem in R N for the parabolic equation ut + div F (u) = ∆ϕ(u), which can degenerate into a hyperbolic equation for some intervals of values of u. In the context of conservation laws (the case ϕ ≡ 0), it is known that an entropy solution can be non-unique when F has singularities. We show the uniqueness of an entropy solution to the general parabolic problem for all L ∞ initial datum, under the isotropic condition on the flux F known for conservation laws. The only assumption on the diffusion term is that ϕ is a non-decreasing continuous function.