Related papers: Fundamental solution and diffusion limits for the …
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of…
This paper deals with the heat equation posed in a bounded regular domain coupled with a dynamical boundary condition of reactive-diffusive type, involving the Laplace-Beltrami operator. We prove well-posedness of the problem together with…
Recently, we have proposed a new free boundary problem representing the bread baking process in a hot oven. Unknown functions in this problem are the position of the evaporation front, the temperature field and the water content. For…
We obtain uniform in time bounds for the solutions to a class of thermo-diffusive systems with classical and fractional diffusions. In the classical diffusion case, the nonlinearities are assumed to be at most exponentially growing, while…
We derive the hydrodynamic limit of a kinetic equation with a stochastic, short range perturbation of the velocity operator. Under some mixing hypotheses on the stochastic perturbation, we establish a diffusion-approximation result: the…
We investigate the inverse problem of numerically identifying unknown initial temperatures in a heat equation with dynamic boundary conditions whenever some overdetermination data is provided after a final time. This is a backward parabolic…
We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is…
We study the asymptotic behaviour of sharp front solutions arising from the nonlinear diffusion equation \theta_t = (D(\theta)\theta_x)_x, where the diffusivity is an exponential function D({\theta}) = D_o exp(\beta\theta). This problem…
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and…
This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)\Delta U(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive.…
A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form, and derive the explicit expression…
In this paper, the existence, uniqueness and dependence on initial value of solution for a singular diffusion equation with nonlinear boundary condition are discussed. It is proved that there exists a unique global smooth solution which…
In this paper, the authors consider an initial boundary value problem for the heat flow of equation of surfaces with constant mean curvatures, which was investigated in [On the heat flow of equation of surfaces of constant mean curvatures,…
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey…
We prove that the initial-value problem for the fractional heat equation admits a solution provided that the (possibly unbounded) initial datum has a conveniently moderate growth at infinity. Under the same growth condition we also prove…
We provide a mathematical analysis of the `diffusion-free' boundary conditions recently introduced by Lin and Kerswell for the numerical treatment of inertial waves in a fluid contained in a rotating sphere. We consider here the full…
We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in $\mathbb{R}^d$. An important special case is the time-fractional diffusion equation, which has seen much…
We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions…
In this work, we consider a number of boundary-value problems for time-fractional heat equation with the recently introduced Caputo-Fabrizio derivative. Using the method of separation of variables, we prove a unique solvability of the…