We give asymptotic formulae for the propagation of an initial disturbance of the Burgers’ equation.

The paper concerns the (local and global) existence, nonexistence, uniqueness and some properties of nonnegative solutions of a nonlinear density dependent diffusion equation with homogeneous Dirichlet boundary conditions.

We consider the homogeneous Schrödinger equation with a long-range potential and show that its solutions satisfying some a priori bound at infinity can asymptotically be expressed as a sum of incoming and outgoing distorted spherical waves. Coefficients of these waves are related by the scattering matrix. This generalizes a similar result obtained earlier in the short-range case.

We study the asymptotic behaviour near infinity of the weak solutions of the Cauchy-problem.

We study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-Gérard [2] on ${ℝ}^3$, is performed by taking care of possible...

We consider the linear convection-diffusion equation associated to higher order elliptic operators⎧  ut + Ltu = a∇u   on Rnx(0,∞)⎩  u(0) = u0 ∈ L1(Rn),where a is a constant vector in Rn, m ∈ N*, n ≥ 1 and L0 belongs to a class of higher order elliptic operators in divergence form associated to non-smooth bounded measurable coefficients on Rn. The aim of this paper is to study the asymptotic behavior, in Lp (1 ≤ p ≤ ∞), of the derivatives Dγu(t) of the solution of the convection-diffusion equation...

We consider the second order parabolic partial differential equation    ${\sum }_{i,j=1}^n{a}_{ij}(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i(x,t)u_{x_i}+c(x,t)u-u_t=0$. Sufficient conditions are given under which every solution of the above equation must decay or tend to infinity as |x|→ ∞. A sufficient condition is also given under which every solution of a system of the form    $L^{\alpha }\left[u^{\alpha }\right]+{\sum }_{\beta =1}^N{c}^{\alpha \beta }(x,t)u^{\beta }=f^{\alpha }(x,t)$, where    $L^{\alpha }\left[u\right]\equiv {\sum }_{i,j=1}^n{a}_{ij}^{\alpha }(x,t)u_{x_i{x}_j}+{\sum }_{i=1}^n{b}_i^{\alpha }(x,t)u_{x_i}-u_t$, must decay as t → ∞.

In this paper, for the second initial boundary value problem for Schrödinger systems, we obtain a performance of generalized solutions in a neighborhood of conical points on the boundary of the base of infinite cylinders. The main result are asymptotic formulas for generalized solutions in case the associated spectrum problem has more than one eigenvalue in the strip considered.

The paper deals with the asymptotic behavior of generalized solutions to nonlinear first order equations. With the aid of explicit variational representation one studies the decrease of solutions for a large time. And for the small time an asymptotic of the perturbation?s front is calculated.