Related papers: An inhomogeneous porous medium equation with large…
We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0, u_1) whose energy is smaller than that of (W,0), where W is the well-known radial positive solution to the corresponding ellipyic equation, the following dichotomy…
In this paper we consider a porous-elastic system consisting of nonlinear boundary/interior damping and nonlinear boundary/interior sources. Our interest lies in the theoretical understanding of the existence, finite time blow-up of…
We show the existence of self-similar solutions with constant finite mass to the time-fractional Porous-Medium Equation for all spatial dimensions $d \ge 1$ and all exponents $m>m_c=(d-2)_+/d$. This range is optimal. We find two types of…
This paper studies a discrete-time version of the Lucas-Uzawa endogenous growth model with physical and human capital in the presence of externalities. Existence of an optimal equilibrium is proved using tools from dynamic programming with…
We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth…
We consider a possibly degenerate porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated…
We show the well-posedness for a large class of degenerate parabolic equations with an additional singularity and mixed Dirichlet-Neumann boundary conditions on bounded Lipschitz domains. The proof is based on an $L^1$-contraction result.…
We study a kind of generalized porous medium equation with fractional Laplacian and abstract pressure term. For a large class of equations corresponding to the form: $u_t+\nu \Lambda^{\beta}u=\nabla\cdot(u\nabla Pu)$, we get their local…
We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under…
We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we…
We study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains $\Omega\subseteq{\mathbb R}^N$, $N\ge3$, with homogeneous Neumann boundary conditions. Firstly we improve some known results in…
In this paper, we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain: {equation} {cases} u_t=\La u^m\quad\text{in $\Omega\times [0,\infty)$} u(x,0)=u_0>0\quad\text{in…
We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan-Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as…
We consider the spatially inhomogeneous Boltzmann equation without angular cutoff for soft potentials. For any given initial datum such that the mass, energy and entropy densities are bounded and the mass is away from vacuum, we establish…
We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_{t} - {\rm div} \left( m |u|^{m-1} \nabla u \right) = f \in L^{q,r}, \quad m >1 ,$$ are locally H\"older continuous, with exponent $$\gamma =\min \left\{…
On the three dimensional Euclidean space, for data with finite energy, it is well-known that the Maxwell-Klein-Gordon equations admit global solutions. However, the asymptotic behaviours of the solutions for the data with non-vanishing…
We consider the inhomogeneous biharmonic nonlinear Schr\"odinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +\Delta^2 u -|x|^{-b} |u|^{2\sigma}u = 0,$$ where $\sigma>0$ and $b>0$. We first study the local well-posedness in $\dot…
We consider a special class of infinite energy solutions to the inviscid incompressible porous medium equations (IPM), introduced in Castro-C\'ordoba-Gancedo-Orive [9]. The (IPM) equations then reduce to a one-dimensional nonlocal nonlinear…
We study the Boltzmann equation for a space-homogeneous gas of inelastic hard spheres, with a diffusive term representing a random background forcing. Under the assumption that the initial datum is a nonnegative $L^2$ function, with bounded…
We consider non-gauge-invariant cubic nonlinear Schr\"odinger equations in one space dimension. We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^\infty$ decay up to time…