Asymptotic Analysis

We prove that for any global solution to the Vlasov–Maxwell system arising from compactly supported data, and such that the electromagnetic field decays fast enough, the distribution function exhibits a modified scattering dynamic. In particular, our result applies to small data solutions constructed by Glassey and Strauss.

In this work, we study the well-posedness and the asymptotic behavior of a thermoelastic shear beam model, where the heat conduction is given by the Gurtin–Pipkin law acting on the bending moment under Dirichlet–Neumann–Dirichlet boundary conditions. To establish the existence and uniqueness of the solution, we use the semigroup method. Then, we show that the system is not exponentially stable. Ultimately, we prove that the system is polynomially stable.

We establish pointwise asymptotic estimates at infinity for solutions to the doubly weighted quasi-linear equation: { − div ( w 1 | ∇ u | p − 2 ∇ u ) = w 2 | u | q − 2 u in  Ω , u ∈ D 1 , p , w 1 ( Ω ) , where w 1 and w 2 are compatible weights and q > p > 1 is a critical exponent in the sense of Sobolev.