Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userHyperbolic Relaxation of Reaction Diffusion Equations with Dynamic Boundary Conditions
104
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Under consideration is the hyperbolic relaxation of a semilinear reaction-diffusion equation on a bounded domain, subject to a dynamic boundary condition. We also consider the limit parabolic problem with the same dynamic boundary condition. Each problem is well-posed in a suitable phase space where the global weak solutions generate a Lipschitz continuous semiflow which admits a bounded absorbing set. We prove the existence of a family of global attractors of optimal regularity. After fitting both problems into a common framework, a proof of the upper-semicontinuity of the family of global attractors is given as the relaxation parameter goes to zero. Finally, we also establish the existence of exponential attractors.
rate research
Read More
We establish an integral variational principle for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions, for arbitrary reaction terms. This principle allows to obtain in a simple way the dependence of the speed on the Stefan constant. As an application a generalized Zeldovich-Frank-Kamenetskii lower bound for the speed, valid for monostable and combustion reaction terms, is given.
We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion-reaction equations with gradient-dependent diffusivity using the two-step version of the method of furcate splitting. For simultaneously finding the equivalence groups of an unnormalized class of differential equations and a collection of its subclasses, we suggest an optimized version of the direct method. The optimization includes the preliminary study of admissible transformations within the entire class and the successive splitting of the corresponding determining equations with respect to arbitrary elements and their derivatives depending on auxiliary constraints associated with each of required subclasses. In the course of applying the suggested technique to subclasses of the class under consideration, we construct, for the first time, a nontrivial example of finite-dimensional effective generalized equivalence group. Using the method of Lie reduction and the generalized separation of variables, exact solutions of some equations under consideration are found.
We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. The model is equipped with generalized Wentzell-Robin (or dynamic) boundary conditions. This allows modelling of adhesion at extremely small or large sizes. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. This is carried out via establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is our model admits a finite dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.
We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.
Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism is necessary, using, for instance, impulsive reaction-diffusion equations. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction-diffusion equations. First, they deal with the existence of traveling waves for monotone systems of impulsive reaction-diffusion equations. Second, they allow the computation of spreading speeds for monotone systems of impulsive reaction-diffusion equations. We apply our methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
