Related papers: Cauchy problem for semilinear wave equation with t…

Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad &…

We consider the Cauchy problem for the wave equation in $\Omega\times{\mathbb R}$ with data given on some part of the boundary $\partial\Omega\times{\mathbb R}$. We provide a reconstruction algorithm for this problem based on analytic…

We consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_\xi f(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} with $q \in (0, \infty)$ in a…

Firstly, we study the equation $\square u = |u|^{q_c}+ |\partial u|^p$ with small data, where $q_c$ is the critical power of Strauss conjecture and $p\geq q_c.$ We obtain the optimal lifespan…

We analyze systems of semilinear wave equations in $3+1$ dimensions whose associated asymptotic equation admit bounded solutions for suitably small choices of initial data. Under this special case of the weak null condition, which we refer…

We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schr\"odinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness…

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped $\sigma$-evolution models with different power nonlinearities. By assuming additional $L^m$ regularity on the initial data, with $m…

We study the inverse problem of recovery a non-linearity $f(x,u)$, which is compactly supported in $x$, in the semilinear wave equation $u_{tt}-\Delta u+ f(x,u)=0$. We probe the medium with either complex or real-valued harmonic waves of…

The subject of the paper is the Cauchy problem for the wave equation in a space-time cylinder $\Omega\times{\mathbb R}$, $\Omega\subset{\mathbb R}^2$, with the data on the surface $\partial\Omega\times I$, where $I$ is a finite time…

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on $(R^3, g)$, where the metric $g$ is a small perturbation of the flat metric and approaches the…

In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) =…

We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the…

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. As is pointed out by [8], in this combination, the frictional damping term is dominant for the viscoelastic one for the…

In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient $g=g(t)$: \begin{equation} \label{EqAbstract} \tag{$\star$} \begin{cases} u_{tt}- \Delta u +…

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho…

In this paper, we consider the following Cauchy problem of a weighted gradient system of semilinear wave equations \begin{equation*} \left\{ \begin{array}{lll} u_{tt}-\Delta u=\lambda |u|^{\alpha}|v|^{\beta+2}u,\quad v_{tt}-\Delta v=\mu…

In this paper we consider a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space with $p \in [3,5)$. We prove that if initial data $(u_0, u_1)$ are radial so that…

Global existence for small data Cauchy problem of semilinear wave equations with scaling invariant damping in 3-D is established in this work, assuming that the data are radial and the constant in front of the damping belongs to $[1.5, 2)$.…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…

We study the Cauchy problem for the two-component Novikov system with initial data $u_0, v_0$ in $H^1(\mathbb{R})$ such that the product $(\partial_x u_0)\partial_x v_0$ belongs to $L^2(\mathbb{R})$. We construct a global semigroup of…