Related papers: Diffusion over ramified domains: solvability and f…

This paper investigates the convergence rate for Tikhonov regularization of the problem of identifying the coefficient $a \in L^{\infty}(\Omega)$ in the Robin-boundary equation $-\mathrm{div}(a\nabla u)-bu=f,~ x \in \Omega \subset \mathbb…

We propose a boundary integral formulation for the dynamic problem of electromagnetic scattering and transmission by homogeneous dielectric obstacles. In the spirit of Costabel and Stephan, we use the transmission conditions to reduce the…

Let $n\ge2$ and $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations…

We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the…

This work considers a model for oncolytic virotherapy, as given by the reaction-diffusion-taxis system $$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v)-\rho uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = D_w \Delta w - w +…

We establish a new $W^{1,2\frac{n-1}{n-2}}$ estimate for the extremal solution of $-\Delta u=\lambda f(u)$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$, which is convex, for arbitrary positive and increasing nonlinearities $f\in…

The main objective of this paper is analysis of the initial-boundary value problems for the linear and semilinear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo…

We consider a diffusion and a wave equations: $$ \partial_t^ku(x,t) = \Delta u(x,t) + \mu(t)f(x), \quad x\in \Omega, \, t>0, \quad k=1,2 $$ with the zero initial and boundary conditions, where $\Omega \subset \mathbb{R}^d$ is a bounded…

We consider the nonlinear Schr\"{o}dinger equation $-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in…

Motivated by the numerical investigation by Aoki et al. [1], we study a rarefied gas flow between two parallel infinite plates of the same temperature governed by the Boltzmann equation with diffuse reflection boundaries, where one plate is…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $i_0\in\mathbb{Z}^+$, $\Omega\subset\mathbb{R}^n$ be a smooth bounded domain, $a_1,a_2,\dots,a_{i_0}\in\Omega$, $\widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\}$, $0\le f\in…

In this paper, we deal with analysis of the initial-boundary value problems for the semilinear time-fractional diffusion equations, while the case of the linear equations was considered in the first part of the present work. These equations…

In this paper, we consider the stochastic reaction-diffusion equation $\mathrm{d}u = (\mathcal{A} u + f(u))\mathrm{d}t + \sigma(u)\mathrm{d}W$ on a smooth bounded domain $\mathcal{O}$ with homogeneous Dirichlet boundary conditions. We…

We study the uniqueness and expansion properties of the positive solution of the logistic equation $\Delta u+au=b(x)f(u)$ in a smooth bounded domain $\Omega$, subject to the singular boundary condition $u=+\infty$ on $\partial\Omega$. The…

The Jordan-Moore-Gibson-Thompson equation \[ \tau u_{ttt} + \alpha u_{tt} = \beta \Delta u_t + \gamma \Delta u + (f(u))_{tt} \] is considered in a smoothly bounded domain $\Omega \subset\mathbb{R}^n$ with $n\leq 3$, where…

This paper is devoted to the investigation of propagation of singularities in hyperbolic equations with non-smooth oefficients, using the Colombeau theory of generalized functions. As a model problem, we study the Cauchy problem for the…

This paper is concerned with a class of reaction-diffusion system with density-suppressed motility \begin{equation*} \begin{cases} u_{t}=\Delta(\gamma(v) u)+\alpha u F(w), & x \in \Omega, \quad t>0, \\ v_{t}=D \Delta v+u-v, & x \in \Omega,…

In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem \begin{equation*} \left\{\begin{array}{rcl} \partial_t u &=& \Delta u - u +…

Diffusion of a penetrating liquid in a polymeric material does not often satisfy the classical diffusion equations and requires taking relaxational (viscoelastic) properties of the polymer into account. We investigate a boundary value…

A classical regularity result is that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We…