This work is concerned with the proof of $L_p-L_q$ decay estimates for solutions of the Cauchy problem for the Klein-Gordon type equation $u_{tt}-{\lambda }^2\left(t\right)b^2\left(t\right)(\Delta u-m^{2}u)=0$. The coefficient consists of an increasing smooth function $\lambda$ and an oscillating smooth and bounded function b which are uniformly separated from zero. Moreover, $m^2$ is a positive constant. We study under which assumptions for λ and b one can expect as an essential part of the decay rate the classical Klein-Gordon decay rate n/2(1/p-1/q).

We establish an upper bound on the Kolmogorov’s entropy of the locally compact attractor for strongly damped wave equation posed in locally uniform spaces in subcritical case using the method of trajectories.

If $\Omega \subset {ℝ}^n$ is a bounded domain with Lipschitz boundary $\partial \Omega$ and $\Gamma$ is an open subset of $\partial \Omega$, we prove that the following inequality $${\left({\int }_{\Omega }{\left|A\left(x\right)\nabla u\left(x\right)\right|}^{p}dx\right)}^{1/p}+{\left({\int }_{\Gamma }{\left|u\left(x\right)\right|}^{p}d{\mathscr{H}}^{n-1}\left(x\right)\right)}^{1/p}\ge c{\parallel u\parallel }_{W^{1,p}\left(\Omega \right)}$$ holds for all $u\in W^{1,p}(\Omega ;{ℝ}^m)$ and $1<p<\infty$, where $${\left(A\left(x\right)\nabla u\left(x\right)\right)}_k=\sum _{i=1}^m\sum _{j=1}^n{a}_k^{ij}\left(x\right)\frac{\partial u_i}{\partial x_j}\left(x\right)(k=1,2,...,r;r\ge m)$$ defines an elliptic differential operator of first order with continuous coefficients on $\overline{\Omega }$. As a special case we obtain $${\int }_{\Omega }{\left|\nabla u\left(x\right)F\left(x\right)+{\left(\nabla u\left(x\right)F\left(x\right)\right)}^T\right|}^{p}dx\ge c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^{p}dx,(*)$$ for all $u\in W^{1,p}(\Omega ;{ℝ}^n)$ vanishing on $\Gamma$, where $F:\overline{\Omega }\to M^{n\times n}\left(ℝ\right)$ is a continuous mapping with $detF\left(x\right)\ge \mu >0$. Next we show that $(*)$ is not valid if $n\ge 3$, $F\in L^{\infty }\left(\Omega \right)$ and $detF\left(x\right)=1$, but does hold if $p=2$, $\Gamma =\partial \Omega$ and $F\left(x\right)$ is symmetric and positive definite in $\Omega$.

The aim of this note is to give a short review of our recent work (see [5]) with Miguel A. Alejo and Luis Vega, concerning the $L^2$-stability, and asymptotic stability, of the $N$-soliton of the Korteweg-de Vries (KdV) equation.

The $L^2$-stability (instability) of a binary nonlinear reaction diffusion system of P.D.E.s - either under Dirichlet or Neumann boundary data - is considered. Conditions allowing the reduction to a stability (instability) problem for a linear binary system of O.D.E.s are furnished. A peculiar Liapunov functional $V$ linked (together with the time derivative along the solutions) by direct simple relations to the eigenvalues, is used.

We prove existence of weak solutions to nonlinear parabolic systems with p-Laplacians terms in the principal part. Next, in the case of diagonal systems an $L_{\infty }$-estimate for weak solutions is shown under additional restrictive growth conditions. Finally, $L_{\infty }$-estimates for weakly nondiagonal systems (where nondiagonal elements are absorbed by diagonal ones) are proved. The $L_{\infty }$-estimates are obtained by the Di Benedetto methods.

In this paper, we consider the global existence, uniqueness and $L^{\infty }$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type $u_t-{\mathrm{div}\left(\right|\nabla u|}^{m-2}{\nabla u)=u|u|}^{\beta -1}{\int }_{\Omega }{\left|u\right|}^{\alpha }\mathrm{d}x$ in $\Omega \times (0,\infty )$ with zero Dirichlet boundary condition in $\partial \Omega$. Further, we obtain the $L^{\infty }$ estimate of the solution $u\left(t\right)$ and $\nabla u\left(t\right)$ for $t>0$ with the initial data $u_0\in L^q\left(\Omega \right)$$(q>...$

We study the decay in time of the spatial $L^p$-norm (1 ≤ p ≤ ∞) of solutions to parabolic conservation laws with dispersive and dissipative terms added uₜ - uₓₓₜ - νuₓₓ + buₓ = f(u)ₓ or uₜ + uₓₓₓ - νuₓₓ + buₓ = f(u)ₓ, and we show that under general assumptions about the nonlinearity, solutions of the nonlinear equations have the same long time behavior as their linearizations at the zero solution.

In this Note we give $L^p$ estimates $\left(1<p<+\mathrm{\infty }\right)$ for the highest order derivatives of an elliptic system in non-divergence form with coefficients in VMO.