The propagation phenomenon of solutions of a parabolic problem on the sphere

2018 ◽  
Vol 28 (10) ◽  
pp. 2001-2067 ◽  
Author(s):  
Bogdan Kazmierczak ◽  
Je-Chiang Tsai ◽  
Slawomir Bialecki
Keyword(s):  
Reaction Kinetics ◽  
A Priori Estimates ◽  
Mild Solutions ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Discontinuity Point ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Uniform Estimates ◽  
Propagation Of Waves

In this paper, we study propagation phenomena on the sphere using the bistable reaction–diffusion formulation. This study is motivated by the propagation of waves of calcium concentrations observed on the surface of oocytes, and the propagation of waves of kinase concentrations on the B-cell membrane in the immune system. To this end, we first study the existence and uniqueness of mild solutions for a parabolic initial-boundary value problem on the sphere with discontinuous bistable nonlinearities. Due to the discontinuous nature of reaction kinetics, the standard theories cannot be applied to the underlying equation to obtain the existence of solutions. To overcome this difficulty, we give uniform estimates on the Legendre coefficients of the composition function of the reaction kinetics function and the solution, and a priori estimates on the solution, and then, through the iteration scheme, we can deduce the existence and related properties of solutions. In particular, we prove that the constructed solutions are of [Formula: see text] class everywhere away from the discontinuity point of the reaction term. Next, we apply this existence result to study the propagation phenomenon on the sphere. Specifically, we use stationary solutions and their variants to construct a pair of time-dependent super/sub-solutions with different moving speeds. When applied to the case of sufficiently small diffusivity, this allows us to infer that if the initial concentration of the species is above the inhomogeneous steady state, then the species will exhibit the propagating behavior.

FINITE DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED PARABOLIC EQUATIONS IN THE PRESENCE OF INITIAL AND BOUNDARY LAYERS

2008 ◽  
Vol 13 (4) ◽  
pp. 483-492
Author(s):  
Nicolas Cordero ◽  
Kevin Cronin ◽  
Grigorii Shishkin ◽  
Lida Shishkina ◽  
Martin Stynes
Keyword(s):  
Difference Scheme ◽  
Boundary Layers ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Parabolic Problems

The grid approximation of an initial‐boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation. The second‐order spatial derivative and the temporal derivative in the differential equation are multiplied by parameters å 2 1 and å 2 2, respectively, that take arbitrary values in the open‐closed interval (0,1]. The solutions of such parabolic problems typically have boundary, initial layers and/or initial‐boundary layers. A priori estimates are constructed for the regular and singular components of the solution. Using such estimates and the condensing mesh technique for a tensor‐product grid, piecewise‐uniform in xand t, a difference scheme is constructed that converges å‐uniformly at the rate O(N−2 ln2 N + N0 −1 ln N0 ), where (N + 1) and (N0 + 1) are the numbers of mesh points in x and t respectively.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

2002 ◽  
Vol 13 (3) ◽  
pp. 337-351 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
C. V. NIKOLOPOULOS ◽  
D. E. TZANETIS
Keyword(s):  
Blow Up ◽  
Ohmic Heating ◽  
Numerical Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Upper And Lower Bounds ◽  
Heating Process ◽  
Non Local ◽  
Initial Boundary Value

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

scholarly journals Generation of upstream advancing solitons by moving disturbances

1987 ◽  
Vol 184 ◽  
pp. 75-99 ◽  
Author(s):  
T. Yao-Tsu Wu
Keyword(s):  
Solitary Waves ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Theoretical Models ◽  
Computer Code ◽  
Basic Mechanism ◽  
Stationary Solutions ◽  
Boussinesq Model ◽  
Dispersive Waves ◽  
Fkdv Equation

This study investigates the recently identified phenomenon whereby a forcing disturbance moving steadily with a transcritical velocity in shallow water can generate, periodically, a succession of solitary waves, advancing upstream of the disturbance in procession, while a train of weakly nonlinear and weakly dispersive waves develops downstream of a region of depressed water surface trailing just behind the disturbance. This phenomenon was numerically discovered by Wu & Wu (1982) based on the generalized Boussinesq model for describing two-dimensional long waves generated by moving surface pressure or topography. In a joint theoretical and experimental study, Lee (1985) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced Korteweg-de Vries (fKdV) equations, both containing forcing functions. The fKdV model is applied in the present study to explore the basic mechanism underlying the phenomenon.To facilitate the analysis of the stability of solutions of the initial-boundary-value problem of the fKdV equation, a family of forced steady solitary waves is found. Any such solution, if once established, will remain permanent in form in accordance with the uniqueness theorem shown here. One of the simplest of the stationary solutions, which is a one-parameter family and can be scaled into a universal similarity form, is chosen for stability calculations. As a test of the computer code, the initially established stationary solution is found to be numerically permanent in form with fractional uncertainties of less than 2% after the wave has traversed, under forcing, the distance of 600 water depths. The other numerical results show that when the wave is initially so disturbed as to have to rise from the rest state, which is taken as the initial value, the same phenomenon of the generation of upstream-advancing solitons is found to appear, with a definite time period of generation. The result for this similarity family shows that the period of generation, Ts, and the scaled amplitude α of the solitons so generated are related by the formula Ts = const α−3/2. This relation is further found to be in good agreement with the first-principle prediction derived here based on mass, momentum and energy considerations of the fKdV equation.

Stability of a weak solution for a hyperbolic system with distributed parameters on a graph

2021 ◽  
pp. 55-67
Author(s):  
Vyacheslav V. Provotorov ◽  
Alexei P. Zhabko
Keyword(s):  
Weak Solution ◽  
Hyperbolic System ◽  
A Priori Estimates ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Summable Function ◽  
Weak Formulation ◽  
Weakly Compact ◽  
Generalized Fourier Series ◽  
Distributed Parameters

In the work, the stability conditions for a solution of an evolutionary hyperbolic system with distributed parameters on a graph describing the oscillating process of continuous medium in a spatial network are indicated. The hyperbolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral identity which determines the variational formulation for the initial-boundary value problem. The basic idea, that has determined the content of this work, is to present a weak solution in the form of a generalized Fourier series and continue with an analysis of the convergence of this series and the series obtained by its single termwise differentiation. The used approach is based on a priori estimates of a weak solution and the construction (by the Fayedo–Galerkin method with a special basis, the system of eigenfunctions of the elliptic operator of a hyperbolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems of oscillations of netset-like industrial constructions which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

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.

scholarly journals Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-16
Author(s):  
Mark A. McKibben
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fixed Point Theory ◽  
Mild Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Point Theory ◽  
Transverse Motion ◽  
Probability Measures ◽  
Uniqueness Results

We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model.

An investigation of the Green–Lindsay three-dimensional model

2017 ◽  
Vol 23 (7) ◽  
pp. 987-1003 ◽  
Author(s):  
Gia Avalishvili ◽  
Mariam Avalishvili ◽  
Wolfgang H Müller
Keyword(s):  
Boundary Value Problem ◽  
Variational Formulation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Relaxation Times ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Three Dimensional Model ◽  
Initial Boundary Value

In this paper we consider the Green and Lindsay nonclassical model for inhomogeneous anisotropic thermoelastic bodies with two relaxation times, which depend on space variables. We obtain a variational formulation for the initial-boundary value problem corresponding to the Green–Lindsay model. On the basis of the variational formulation we define the spaces of vector-valued distributions corresponding to the initial-boundary value problem and by applying suitable a priori estimates we prove the existence and uniqueness of the solution, an energy equality, and the continuous dependence of the solution on given data.

GLOBAL WEAKLY DISCONTINUOUS SOLUTIONS TO THE MIXED INITIAL–BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

2009 ◽  
Vol 19 (07) ◽  
pp. 1099-1138 ◽  
Author(s):  
ZHI-QIANG SHAO
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Local Existence ◽  
Discontinuous Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Quasilinear Hyperbolic Systems ◽  
Initial Boundary Value

In this paper, we consider the mixed initial–boundary value problem for first-order quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x) | t ≥ 0, x ≥ 0}. Based on the fundamental local existence results and global-in-time a priori estimates, we prove the global existence of a unique weakly discontinuous solution u = u(t, x) with small and decaying initial data, provided that each characteristic with positive velocity is weakly linearly degenerate. Some applications to quasilinear hyperbolic systems arising in physics and other disciplines, particularly to the system describing the motion of the relativistic closed string in the Minkowski space R1+n, are also given.

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