Related papers: Group Analysis of Variable Coefficient Diffusion-C…
We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using…
We explore the advection-diffusion of a passive vector described by $\partial_t u + U \cdot \nabla u = - \nabla p + \nu \Delta u$, where both $U$ and $u$ are divergence-free velocity fields. We approach this equation from an input/output…
In this paper, Sturm-Liouville problem for difference equations is considered with potential function q(n). The representations of solutions are obtained by variation of parameters method. These solutions are proved, using summation by…
The linear response description for impurity diffusion in a granular fluid undergoing homogeneous cooling is developed in the preceeding paper. The formally exact Einstein and Green-Kubo expressions for the self-diffusion coefficient are…
Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the…
We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion $$ \partial_tu+(-\Delta)^{1/2}\log(1+u)=0, $$ posed for $x\in \mathbb{R}$, with nonnegative initial data…
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of H{\"o}lder continuous coefficients as well as piecewise smooth drifts with…
The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular, that the above class contains no…
We study a class of fourth-order quasilinear degenerate parabolic equations under both time-and space-dependent and time-and space-independent forces, modeling non-Newtonian thin-film flow over a solid surface in the "complete wetting"…
The Lie-group approach was applied to determine symmetries of the third-order non-linear equation formulated for description of shear elastic disturbances in soft solids. Invariant solutions to this equation are derived and it turned out…
We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term.…
We examine stochastic reaction-diffusion equations of the form $\frac{\partial u}{\partial t} = \mathcal{A} u(t,x) + f(u(t,x)) + \sigma(u(t,x))\dot{W}(t,x)$ and provide sufficient conditions on the reaction term and multiplicative noise…
We prove two stability results for the scale invariant solutions of the nonlinear heat equation $\partial_t u=\Delta u - |u|^{p-1}u$ with $1<p<1+{2\over n}$, $n$ being the spatial dimension. The first result is that a small perturbation of…
The paper considers the wave equation, with constant or variable coefficients in $\R^n$, with odd $n\geq 3$. We study the asymptotics of the distribution $\mu_t$ of the random solution at time $t\in\R$ as $t\to\infty$. It is assumed that…
A new transformation for radially symmetric solutions to the subcritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\mathbb{R}^N\times(0,\infty)$ and with…
The~numerical solutions to a non-linear Fractional Fokker--Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The~aim is to model anomalous diffusion using an FFP description with fractional velocity…
Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…
The purpose of this work is to analyze the blow-up of solutions of the nonlinear parabolic equation \[ u_t-\Delta u=|x|^{\alpha}|u|^{p}+{\mathtt a}(t)\textbf{w}(x) \ \quad\mbox{for } (t,x)\in(0,\infty)\times\mathbb{R}^{N}, \] where $p>1$,…
We consider similarity solutions of the generalized convection-diffusion-reaction equation with both space- and time-dependent convection, diffusion and reaction terms. By introducing the similarity variable, the reaction-diffusion equation…
We study the Cauchy problem for the advection-diffusion equation $\partial_t u + \mathrm{div} (u b ) = \Delta u$ associated with a merely integrable divergence-free vector field $b$ defined on the torus. We discuss existence, regularity and…