Exponential Growth of Solution for a Class of Reaction Diffusion Equation with Memory and Multiple Nonlinearities

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

We prove the interior approximate controllability of the following broad class of reaction diffusion equation in the Hilbert spaces Z = L2(Ω) given by z′ = −Az + 1ωu(t), t ∈ [0, τ], where Ω is a domain in ℝn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u ∈ L2(0, t1; L2(Ω)) and A : D(A) ⊂ Z → Z is an unbounded linear operator with the following spectral decomposition: 〈z, ϕj,k〉ϕj,k. The eigenvalues 0 < λ1 < λ2 < ⋯<⋯λn → ∞ of A have finite multiplicity γj equal to the dimension of the corresponding eigenspace, and {ϕj,k} is a complete orthonormal set of eigenvectors of A. The operator −A generates a strongly continuous semigroup {T(t)} given by 〈z, ϕj,k〉ϕj,k. Our result can be applied to the nD heat equation, the Ornstein‐Uhlenbeck equation, the Laguerre equation, and the Jacobi equation.

In this paper, the blow-up of solution for the initial boundary value problem of a class of reaction diffusion equations with multiple nonlinearities is studied. We prove, under suitable conditions on memory and nonlinearities term and for negative or positive initial energy, a global nonexistence theorem.

A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.

The singularly perturbed initial boundary value problem for a class of reaction diffusion equation is considered. Under appropriate conditions, the existence-uniqueness and the asymptotic behavior of the solution are showed by using the fixed-point theorem.

Finite difference solutions of reaction diffusion equations with continuous time delays

Spreading speeds and uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delays

We are concerned with a class of reaction diffusion equations with nonlinear terms of arbitrary growth on unbounded domains. The existence of an $L^2 - L^{2p-2} \cap H^2$ global attractor is proved. This improves the results in previous references, and the proof is shorter.

We study a general class of scalar reaction/interacting population diffusion equations in two space dimensions: convective terms, due to wind, are included. We consider boundary conditions which include a measure of the hostility to the species in the exterior of the domain. The main point of the paper is to obtain estimates for the minimum domain size which can sustain spatially heterogeneous structures and indicate the type of patterns which could appear.

Modelling dispersal of populations and genetic information by finite element methods

Stabilization to a trajectory for the monodomain equations, a coupled nonlinear PDE-ODE system, is investigated. The results rely on stabilization of linear first-order in time nonautonomous evolution equations combined with stabilizability results for the linearized monodomain equations and a fixed point argument to treat local stabilizability of the nonlinear system. Numerical experiments for feedback stabilization of reentry phenomena are included.

Fractal theory is a branch of nonlinear scientific research, and its research object is the irregular geometric form in nature. On account of the complexity of the fractal set, the traditional Euclidean dimension is no longer applicable and the measurement method of fractal dimension is required. In the numerous fractal dimension definitions, box-counting dimension is taken to characterize the complexity of Julia set since the calculation of box-counting dimension is relatively achievable. In this paper, the Julia set of Brusselator model which is a class of reaction diffusion equations from the viewpoint of fractal dynamics is discussed, and the control of the Julia set is researched by feedback control method, optimal control method, and gradient control method, respectively. Meanwhile, we calculate the box-counting dimension of the Julia set of controlled Brusselator model in each control method, which is used to describe the complexity of the controlled Julia set and the system. Ultimately we demonstrate the effectiveness of each control method.

The authors provide a complete classification of the radial solutions to a class of reaction diffusion equations arising in the study of thermal structures such as plasmas with thermal equilibrium or no flux at the boundary. In particular, their study includes rapidly growing nonlinearities, that is, those where an exponent exceeds the critical exponent. They describe the corresponding bifurcation diagrams and determine existence and uniqueness of ground states, which play a central role in characterizing those diagrams. They also provide information on the stability-unstability of the radial steady states.

We study large deviations in the Langevin dynamics, with damping of order $\e^{-1}$ and noise of order $1$, as $\e\downarrow 0$. The damping coefficient is assumed to be state dependent. We proceed first with a change of time and then, we use a weak convergence approach to large deviations and their equivalent formulation in terms of the Laplace principle, to determine the good action functional. Some applications of these results to the exit problem from a domain and to the wave front propagation for a suitable class of reaction diffusion equations are considered.

For a class of reaction diffusion equations in a bounded domain under dissipative dynamical time lateral boundary conditions, the occurence of blow up phenomena is shown by comparison of solutions, as well as by energy and spectral methods. Moreover, the dependence of the blow up time on different boundary conditions is investigated, where the dynamical boundary condition interpolates between the Neumann boundary condition and a certain Dirichlet boundary condition related to the initial condition. Some of the techniques presented here apply also to certain parabolic equations with degenerate principal part. †Dedicated to Professor Helmut Kaul on the occasion of his 65th birthday.