Related papers: p-Adic Analogue of the Porous Medium Equation
We study the Cauchy problem for $p$-adic nonlinear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the…
We develop a theory of generalized solutions of the nonlinear evolution equations for complex-valued functions of a real positive time variable and $p$-adic spatial variable, which can be seen as non-Archimedean counterparts of the…
We study a quite general family of nonlinear evolution equations of diffusive type with nonlocal effects. More precisely, we study porous medium equations with a fractional Laplacian pressure, and the problem is posed on a bounded space…
A subdiffusion problem in which the diffusion term is related to a stable stochastic process is introduced. Linear models of these systems have been studied in a general way, but non-linear models require a more specific analysis. The model…
The final goal of this paper is to prove existence of local (strong) solutions to a (fully nonlinear) porous medium equation with blow-up term and nondecreasing constraint. To this end, the equation, arising in the context of Damage…
In this paper, we prove the existence and the uniqueness of a weak and mild solution of the following nonlinear parabolic problem involving the porous $p$-fractional Laplacian: \begin{equation*} \begin{cases} \partial_t…
We consider nonlinear diffusive evolution equations posed on bounded space domains, governed by fractional Laplace-type operators, and involving porous medium type nonlinearities. We establish existence and uniqueness results in a suitable…
We consider nonlinear nonlocal diffusive evolution equations, governed by fractional Laplace-type operators, fractional time derivative and involving porous medium type nonlinearities. Existence and uniqueness of weak solutions are…
The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative,…
We study nonnegative solutions to the Fractional Porous Medium Equation on a suitable class of connected, noncompact Riemannian manifolds. We provide existence and smoothing estimates for solutions, in an appropriate weak (dual) sense, for…
A change of variables is introduced to reduce certain nonlinear stochastic evolution equations with multiplicative noise to the corresponding deterministic equation. The result is then used to investigate a stochastic porous medium…
In this article, we prove existence and uniqueness of the Barenblatt solution of the evolution equation on the whole Euclidean space where the principle part is the nonlocal fractional p-Laplacian composed with a power function. Our proof…
In this short note we consider a nonlinear and spatially nonlocal PDE modelling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We…
In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic string theory. These equations are of much interest…
Let $\Omega $ be a bounded domain of $\mathbb{R}^{N}(N\geq 2)$. We obtain a necessary and a sufficient condition, expressed in terms of capacities, for existence of a solution to the porous medium equation with absorption \begin{equation*}…
We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak…
We study the Vladimirov fractional differentiation operator $D^\alpha_N$, $\alpha >0, N\in \mathbb Z$, on a $p$-adic ball $B_N=\{ x\in \mathbb Q_p:\ |x|_p\le p^N\}$. To its known interpretations via restriction from a similar operator on…
Consider the non-linear stochastic fractional-diffusion equation \begin{eqnarray*} \left \{\begin{array}{lll} \frac{\partial}{\partial t}u(x,t)= -( \Delta)^{\alpha/2} u^m(x,t) + \sigma(u(x,t)) \dot{W}(x,t),\, x\in \mathbb{R}^d,t>0, u(x,0)=…
The nonnegative solution for a linear degenerate diffusion transport eqution is proved. As a result, we show the existence and uniqueness of the solution for the fractional porous medium equation in Sobolev spaces $H^\alpha$ with…
We study a porous medium equation, with nonlocal diffusion effects given by an inverse fractional Laplacian operator. We pose the problem in n-dimensional space for all t>0 with bounded and compactly supported initial data, and prove…