Positivity for elliptic and parabolic systems

1978 ◽  
Vol 79 (3-4) ◽  
pp. 183-191 ◽  
Author(s):  
J. F. G. Auchmuty
Keyword(s):  
Initial Conditions ◽  
Elliptic Systems ◽  
Reaction Diffusion ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Reaction Diffusion Equations ◽  
Biological Phenomena ◽  
Weakly Coupled ◽  
Coupling Terms ◽  
Initial Boundary Value

SynopsisThe positivity of solutions of initial-boundary value problems for weakly-coupled semilinear parabolic or elliptic systems of equations is studied. Conditions on the coupling terms are described which ensure that the solutions of the parabolic systems remain positive whenever the initial conditions are positive. For elliptic systems involving a parameter, conditions on the coupling terms are described which imply that solution branches which contain a positive solution, in fact, contain only positive solutions. Applications of these theorems to certain reaction-diffusion equations arising in the modelling of biological phenomena are given.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. I. Continuous diffusivity

1995 ◽  
Vol 350 (1694) ◽  
pp. 335-360 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Propagation Speed ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Classical Solutions ◽  
Initial Boundary Value

We examine the effects of a concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics. The diffusivity is taken as a continuous monotone, a decreasing function of concentration that has compact support, of the form that arises in polymerization processes. We consider piecewise-classical solutions to an initial-boundary value problem. The existence of a family of permanent form travelling wave solutions is established, and the development of the solution of the initial-boundary value problem to the travelling wave of minimum propagation speed is considered. It is shown that an interface will always form in finite time, with its initial propagation speed being unbounded. The interface represents the surface of the expanding polymer matrix.

The effects of variable diffusivity on the development of travelling waves in a class of reaction-diffusion equations

1994 ◽  
Vol 348 (1687) ◽  
pp. 229-260 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Travelling Waves ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Classical Solutions ◽  
Initial Boundary Value

In this paper we examine the effects of concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics and ecology. We consider piecewise classical solutions to an initial boundary-value problem. The existence of a family of permanent form travelling wave solutions is established and the development of the solution of the initial boundary-value problem to the travelling wave of minimum propagation speed is considered. For certain types of initial data, ‘waiting time’ phenomena are encountered.

scholarly journals Regularity and Exponential Growth of Pullback Attractors for Semilinear Parabolic Equations Involving the Grushin Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Nguyen Dinh Binh
Keyword(s):  
Exponential Growth ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equations ◽  
Pullback Attractor ◽  
Pullback Attractors ◽  
Grushin Operator ◽  
Initial Boundary Value ◽  
Semilinear Parabolic

Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domainΩ. We prove the regularity and exponential growth of a pullback attractor in the spaceS02(Ω)∩L2p−2(Ω)for the nonautonomous dynamical system associated to the problem. The obtained results seem to be optimal and, in particular, improve and extend some recent results on pullback attractors for reaction-diffusion equations in bounded domains.

scholarly journals NONLINEAR WAVE EQUATIONS AND REACTION–DIFFUSION EQUATIONS WITH SEVERAL NONLINEAR SOURCE TERMS OF DIFFERENT SIGNS AT HIGH ENERGY LEVEL

2013 ◽  
Vol 54 (3) ◽  
pp. 153-170 ◽  
Author(s):  
RUNZHANG XU ◽  
YANBING YANG ◽  
SHAOHUA CHEN ◽  
JIA SU ◽  
JIHONG SHEN ◽  
...  
Keyword(s):  
Boundary Value Problem ◽  
Wave Equations ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Nonlinear Wave Equations ◽  
Initial Boundary Value

AbstractThis paper is concerned with the initial boundary value problem of a class of nonlinear wave equations and reaction–diffusion equations with several nonlinear source terms of different signs. For the initial boundary value problem of the nonlinear wave equations, we derive a blow up result for certain initial data with arbitrary positive initial energy. For the initial boundary value problem of the nonlinear reaction–diffusion equations, we discuss some probabilities of the existence and nonexistence of global solutions and give some sufficient conditions for the global and nonglobal existence of solutions at high initial energy level by employing the comparison principle and variational methods.

Metastable travelling-wave solutions of singularly-perturbed reaction-diffusion equations

1998 ◽  
Vol 9 (4) ◽  
pp. 397-416 ◽  
Author(s):  
JACQUES G. L. LAFORGUE ◽  
ROBERT E. O'MALLEY ◽  
MICHAEL J. WARD
Keyword(s):  
Reaction Diffusion ◽  
Initial Boundary ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Diffusion Equations ◽  
Time Interval ◽  
Long Time ◽  
Spatial Domains ◽  
Initial Boundary Value

This paper determines the asymptotic solution of certain initial-boundary value problems for singularly-perturbed reaction-diffusion equations, including the Allen–Cahn and Cahn–Hilliard equations, on bounded one-dimensional spatial domains for r[ges ]0. Attention is focused on the metastable evolution of a transition layer over an asymptotically exponentially-long time interval.

scholarly journals Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Petr Stehlík ◽  
Jonáš Volek
Keyword(s):  
Maximum Principle ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Maximum Principles ◽  
Time Step ◽  
Reaction Diffusion Model ◽  
Continuous Reaction ◽  
Initial Boundary Value

We study reaction-diffusion equations with a general reaction functionfon one-dimensional lattices with continuous or discrete timeux′  (or  Δtux)=k(ux-1-2ux+ux+1)+f(ux),x∈Z. We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time) exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.

scholarly journals FRACTIONAL APPROXIMATIONS OF THERMOELASTIC PLATE SYSTEMS

2021 ◽  
Author(s):  
Flank Bezerra ◽  
CÁSSIO FEITOSA
Keyword(s):  
Initial Conditions ◽  
Integral Formula ◽  
Geometric Theory ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Semilinear Parabolic Systems ◽  
Thermoelastic Plate ◽  
Initial Boundary Value ◽  
Plate System

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with $N\geqslant 2$, which boundary $\partial\Omega$ is assumed to be a $\mathcal{C}^4$-hypersurface. In this paper we consider the initial-boundary value problem associated with the following thermoelastic plate system \[ \begin{cases} \partial_t^2u +\Delta^2 u+\Delta\theta=f(u),\ & x\in\Omega,\ t>0, \\ \partial_t\theta-\Delta \theta-\Delta \partial_tu=0,\ & x\in\Omega,\ t>0, \end{cases} \] subject to boundary conditions \[ \begin{cases} u=\Delta u=0,\ & x\in\partial\Omega,\ t>0,\\ \theta=0,\ & x\in\partial\Omega,\ t>0, \end{cases} \] and initial conditions \[ u(x,0)=u_0(x),\ \partial_tu(x,0)=v_0(x)\ \mbox{and}\ \theta(x,0)=\theta_0(x),\ x\in\Omega. \] We calculate explicit the fractional powers of the thermoelastic plate operator associated with this system via Balakrishnan integral formula and we present a fractional approximated system. We obtain a result of local well-posedness of the thermoelastic plate system and of its fractional approximations via geometric theory of semilinear parabolic systems.

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