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.

Existence of weak solutions and an $L_{\infty }$-estimate are shown for nonlinear nondegenerate parabolic systems with linear growth conditions with respect to the gradient. The $L_{\infty }$-estimate is proved for equations with coefficients continuous with respect to x and t in the general main part, and for diagonal systems with coefficients satisfying the Carathéodory condition.

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 consider a potential type perturbation of the three dimensional wave equation and we establish a dispersive estimate for the associated propagator. The main estimate is proved under the assumption that the potential $V\ge 0$ satisfies $$\left|V\left(x\right)\right|\le \frac{C}{{\left(1+\left|x\right|\right)}^{2+{ϵ}_0}},$$ where ${ϵ}_0>0$.

Continuity in $L^p$ spaces and spaces of Hölder type is proved for pseudodifferential operators of order zero, under general conditions on the class of symbols. Applications to the regularity theory of some hypoelliptic operators are outlined.

Starting from a general formulation of the characterization by dyadic crowns of Sobolev spaces, the authors give a result of $L^p$ continuity for pseudodifferential operators whose symbol a(x,ξ) is non smooth with respect to x and whose derivatives with respect to ξ have a decay of order ρ with $0<\rho \le 1$. The algebra property for some classes of weighted Sobolev spaces is proved and an application to multi - quasi - elliptic semilinear equations is given.

We prove endpoint estimates for operators given by oscillating spectral multipliers on Riemannian manifolds with $C^{\infty }$-bounded geometry and nonnegative Ricci curvature.

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.

We consider the Schrödinger operators $-\Delta +V\left(x\right)$ in ${ℝ}^n$ where the nonnegative potential $V\left(x\right)$ belongs to the reverse Hölder class $B_q$ for some $q\ge n/2$. We obtain the optimal $L^p$ estimates for the operators $(-\Delta +V{)}^{i\gamma },{\nabla }^2(-\Delta +V{)}^{-1},\nabla (-\Delta +V{)}^{-1/2}$ and $\nabla (-\Delta +V{)}^{-1}$ where $\gamma \in ℝ$. In particular we show that $(-\Delta +V{)}^{i\gamma }$ is a Calderón-Zygmund operator if $V\in B_{n/2}$ and $\nabla (-\Delta +V{)}^{-1/2},\nabla (-\Delta +V{)}^{-1}\nabla$ are Calderón-Zygmund operators if $V\in B_n$.