Related papers: Half-space decay for linear kinetic equations
We consider a model case for a strictly convex domain of dimension $d\geq 2$ with smooth boundary and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay…
A Lagrangian numerical scheme for solving nonlinear degenerate Fokker-Planck equations in space dimensions $d\ge2$ is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies…
We present a series of results focused on the decay in time of solutions of classical and anomalous diffusive equations in a bounded domain. The size of the solution is measured in a Lebesgue space, and the setting comprises time-fractional…
Consider wave equations with time derivative nonlinearity and time-dependent propagation speed which are generalized versions of the wave equations in the Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime, the de Sitter spacetime and…
In this paper, we show that the global solution of the surface anisotropic two-dimensional quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion established by the author in [2] is bounded in…
Based on some elementary estimates for the space-time derivatives of the heat kernel, we use a bootstrapping approach to establish the optimal decay rates for the $L^q(\mathbb{R}^d)$ ($1\leq q\leq\infty$, $d\in\mathbb{N}$) norm of the…
We consider the Cauchy problem for systems of semilinear wave equations in two space dimensions. We present a structural condition on the nonlinearity under which the energy decreases to zero as time tends to infinity if the Cauchy data are…
We investigate the dynamics of roll solutions at the zigzag boundary of the planar Swift-Hohenberg equation. Linear analysis shows an algebraic decay of small perturbation with a $t^{- 1/4}$ rate, instead of the classical $t^{- 1/2}$…
We consider the Zakharov-Kutznesov (ZK) equation posed in $\mathbb R^d$, with $d=2$ and $3$. Both equations are globally well-posed in $L^2(\mathbb R^d)$. In this paper, we prove local energy decay of global solutions: if $u(t)$ is a…
We study existence and long-time behaviour of strong solutions for the thin film equation using a priori estimates in a weighted Sobolev space. This equation can be classified as a doubly degenerate fourth-order parabolic and it models…
We prove pointwise-in-time dispersive decay for solutions to the energy-critical nonlinear Schr\"odinger equation in spatial dimensions $d = 3,4$ for both the initial-value and final-state problems.
We consider degenerate Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative. It is well known that these equations admit global solutions when epsilon is small enough, and that these solutions…
We study steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions and the source term decaying exponentially, we…
We consider a stochastic heat equation of the type, $\partial_t u = \partial^2_x u + \sigma(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $\sigma:\mathbb{R}…
We study the degenerate Kolmogorov equations (also known as kinetic Fokker-Planck equations) in nondivergence form. The leading coefficients $a^{ij}$ are merely measurable in $t$ and satisfy the vanishing mean oscillation (VMO) condition in…
We study the Cauhcy problem for space-time fractional nonlinear Schr\"odinger equation with a general nonlinearity. We prove the local well-posedness of it in fractional Sobolev spaces based on the decay estimates and H\"older type…
In the paper by means of Fourier transform method and similarity method we solve the Dirichlet problem for a multidimensional equation wich is a generalization of the Tricomi, Gellerstedt and Keldysh equations in the half-space, in which…
We obtain estimates on the rate of decay of a solution to the wave equation on a stationary spacetime that tends to Minkowski space at a rate $O(\lvert x \rvert^{-\kappa}),$ $\kappa \in (1,\infty) \backslash \mathbb{N}.$ Given suitably…
We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in $\mathbb R^n$, $n \ge 4$, and deduce the large-time…
We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that…