Related papers: Reaction Diffusion Equations with Nonlinear Bounda…
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 consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary.…
We consider evolutionary reaction-diffusion problem with mixed Dirichlet--Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and exact…
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining…
Deriving evolution equations accounting for both anomalous diffusion and reactions is notoriously difficult, even in the simplest cases. In contrast to normal diffusion, reaction kinetics cannot be incorporated into evolution equations…
Strang splitting is a widely used second-order method for solving diffusion-reaction problems. However, its convergence order is often reduced to order $1$ for Dirichlet boundary conditions and to order $1.5$ for Neumann and Robin boundary…
A popular approach to modeling bimolecular reactions between diffusing molecules is through the use of reactive boundary conditions. One common model is the Smoluchowski partial absorption condition, which uses a Robin boundary condition in…
A kind of nonlocal reaction-diffusion equations on an unbounded domain containing fractional Laplacian operator is analyzed. To be precise, we prove the convergence of solutions of the equation governed by the fractional Laplacian to the…
We consider a nonlinear reaction-diffusion equation settled on the whole euclidean space. We prove the well-posedness of the corresponding Cauchy problem in a general functional setting, namely, when the initial datum is uniformly locally…
Reaction-diffusion models have been used over decades to study biological systems. In this context, evolution equations for probability distribution functions and the associated stochastic differential equations have nowadays become…
The initial inverse problem of finding solutions and their initial values ($t = 0$) appearing in a general class of fractional reaction-diffusion equations from the knowledge of solutions at the final time ($t = T$). Our work focuses on the…
This paper investigates solutions of hyperbolic diffusion equations in $\mathbb{R}^3$ with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere $S^2$ are studied. All…
We study a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)\ (x\in [g(t),h(t)])$ with free boundary conditions $g'(t)=-u_x(t,g(t))+\alpha$ and $h'(t)=-u_x(t,g(t))-\alpha$ for some $\alpha>0$. Such problems may be used to describe…
The question addressed here is the long time evolution of the solutions to a class of one-dimensional reaction-diffusion equations, in which the diffusion is given by an integral operator. The underlying motivation, discussed in the first…
The boundary integral method is extended to derive closed integro-differential equations applicable to computation of the shape and propagation speed of a steadily moving spot and to the analysis of dynamic instabilities in the sharp…
In this work, we investigate a reaction-diffusion system in which both species are influenced by self-diffusion. Due to Hopf's boundary lemma, we obtain the boundedness of the classical solution of the system. By considering a particular…
This paper deals with the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in $\mathbb{R}^N$. We prove a variational formula for the spreading speeds in any direction, and we also…
We have studied the front propagation in a one dimensional case of combustion by solving numerically an advection-reaction-diffusion equation. The physical model is simplified so that no coupling phenomena are considered and the reacting…
This is part II of our study on the free boundary problems with nonlocal and local diffusions. In part I, we obtained the existence, uniqueness, regularity and estimates of global solution. In part II here, we show a spreading-vanishing…
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general…