Related papers: Some $L^1$-$L^1$ estimates for solutions to visco-…

We prove energy estimates for linear $p$-evolution equations in weighted Sobolev spaces under suitable assumptions on the behavior at infinity of the coefficients with respect to the space variables. As a consequence we obtain well…

We consider the Cauchy problem for cubic nonlinear Klein-Gordon equations in one space dimension. We give the $L^p$-decay estimate for the small data solution and show that it decays faster than the free solution if the cubic nonlinearity…

We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of…

The aim of this paper is to derive higher order energy estimates for solutions to the Cauchy problem for damped wave models with time-dependent propagation speed and dissipation. The model of interest is \begin{equation*}…

An initial-boundary value problem for \[ \left\{ \begin{array}{ll} u_{tt} = \big(\gamma(\Theta) u_{xt}\big)_x + au_{xx} - \big(f(\Theta)\big)_x, \qquad & x\in\Omega, \ t>0, \\[1mm] \Theta_t = \Theta_{xx} + \gamma(\Theta) u_{xt}^2 -…

In this paper, we are interested in the Cauchy problem for the viscoelastic damped wave equation with memory of type I. By applying WKB analysis and Fourier analysis, we explain the memory's influence on dissipative structures and…

We consider a viscous approximation for a nonlinear degenerate convection-diffusion equations in two space dimensions, and prove an $L^1$ error estimate. Precisely, we show that the $L^1_{\mathrm{loc}}$ difference between the approximate…

We study the Cauchy problem for the first order evolutive Hamilton-Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and…

In this paper we consider the evolution equation $\partial_t u=\Delta_\mu u+f$ and the corresponding Cauchy problem, where $\Delta_\mu$ represents the Bessel operator $\partial_x^2+(\frac{1}{4}-\mu^2)x^{-2}$, for every $\mu>-1$. We…

We study the Cauchy problem for a class of linear evolution equations of arbitrary order with coefficients depending both on time and space variables. Under suitable decay assumptions on the coefficients of the lower order terms for $|x|$…

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$,…

In this paper we study the long-time stability of the Cauchy one-leg theta-methods for the two-dimensional NavierStokes equations. We establish the uniform dissipativity in H^1, in the sense that the semi-discrete-in-time approximations…

This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional viscous radiative and reactive gas. Based on the elaborate energy estimates, we developed a new approach to derive the upper…

We study the Cauchy problem for the incompressible Navier-Stokes equations (NS) in three and higher spatial dimensions: \begin{align} u_t -\Delta u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0(x). \label{NSa} \end{align}…

Let $G$ be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$. More preciously, we investigate some $L^2$-estimates for the solution to the homogeneous…

We consider the Cauchy problem for the weakly dissipative wave equation $$ \bx v+\frac\mu{1+t}v_t=0, \qquad x\in\R^n,\quad t\ge 0, $$ parameterized by $\mu>0$, and prove a representation theorem for its solution using the theory of special…

This paper studies Hamilton-Jacobi equations of evolution type defined in a general metric space. We give a notion of a solution through optimal principles and establish a unique existence theorem of the solution for initial value problems.…

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-\phi*u), \] where $\phi*u$ is a spatial convolution with the top hat kernel, $\phi(y) \equiv…

By introducing a class of new function spaces $B^{\sigma,s}_{p,q}$ as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon equation (NLKG) in all spatial dimensions $d \geqslant 1$, $$ \partial^2_t u + u- \Delta…

We introduce a new model of the logarithmic type of wave-like equation with a nonlocal logarithmic damping mechanism, which is rather weakly effective as compared with frequently studied fractional damping cases. We consider the Cauchy…