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In this paper, we study the Dirichlet boundary value problem of steady-state relativistic Boltzmann equation in half-line with hard potential model, given the data for the outgoing particles at the boundary and a relativistic global…
Consider a scalar conservation law with discontinuous flux \begin{equation*}\tag{1} \quad u_{t}+f(x,u)_{x}=0, \qquad f(x,u)= \begin{cases} f_l(u)\ &\text{if}\ x<0,\\ f_r(u)\ & \text{if} \ x>0, \end{cases} \end{equation*} where $u=u(x,t)$ is…
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\Delta u+V(x)u)=\epsilon\mu\Delta u+\epsilon \mathcal{P}( u),\quad x\in {R^d}\,, \quad(*) $$ under the periodic boundary conditions, where…
Let $X$ be a smooth $n\,$-dimensional manifold and $D$ be an open connected set in $X$ with smooth boundary $\partial D$. Perturbing the Cauchy problem for an elliptic system $Au = f$ in $D$ with data on a closed set $\iG \subset \partial…
The paper is aimed at analysing a singular perturbation of the Navier-Stokes equations on a compact closed manifold. The case of compact smooth manifolds with boundary under the Dirichlet conditions is also included. Global existence and…
We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic…
Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only H\"older continuous…
We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial…
We study the strong solvability of the Cauchy-Dirichlet problem for parabolic quasilinear equations with discontinuous data. The principal coefficients depend on the point $(x,t)$ and on the solution u, the dependence on x is of VMO type…
We study the Cauchy problem for a coupled system of a complex Ginzburg-Landau equation with a quasilinear conservation law $$ \left\{\begin{array}{rlll} e^{-i\theta}u_t&=&u_{xx}-|u|^2u-\alpha g(v)u& v_t+(f(v))_x&=&\alpha (g'(v)|u|^2)_x&…
In this paper, we study the global existence of solutions of the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations…
The Cauchy problem for the inelastic Boltzmann equation is studied for small data. Existence and uniqueness of mild and weak solutions is obtained for sufficiently small data that lies in the space of functions bounded by Maxwellians. The…
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…
The motion of rarefied gases for uniform shear flow at the kinetic level is governed by the spatially homogeneous Boltzmann equation with a deformation force. In the paper we study the corresponding Cauchy problem with initial data of…
We consider the total energy decay of the Cauchy problem for wave equations with a potential and an effective damping. We treat it in the whole one-dimensional Euclidean space. Fast energy decay is established with the help of potential.…
We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-\Delta)^{\sigma}u+(-\Delta)^{\delta}\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with…
In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. Our aim is to obtain the threshold, to classify the global existence of solution for small data or the finite time…
This paper is concerned with a compressible MHD equations describing the evolution of viscous non-resistive fluids in piecewise regular bounded Lipschitz domains. Under the general inflow-outflow boundary conditions, we prove existence of…
In this paper we study the Boltzmann equation near global Maxwellians in the $d$-dimensional whole space. A unique global-in-time mild solution to the Cauchy problem of the equation is established in a Chemin-Lerner type space with respect…
We study the nonlinear and nonlocal Cauchy problem \[ \partial_{t}u+\mathcal{L}\varphi(u)=0 \quad\text{in }\mathbb{R}^{N}\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0, \] where $\mathcal{L}$ is a L\'evy-type nonlocal operator with a kernel…