scholarly journals A diffusion equation with localized chemical reactions

1994 ◽  
Vol 37 (1) ◽  
pp. 101-118 ◽  
Author(s):  
John M. Chadam ◽  
Hong-Ming Yin
Keyword(s):  
Global Existence ◽  
Diffusion Equation ◽  
Finite Time ◽  
Diffusion Processes ◽  
Model Problem ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Well Posedness ◽  
Blowup Rate ◽  
Time Blowup

In some chemical reaction–diffusion processes, the reaction takes place only at some local sites, due to the presence of a catalyst. In this paper we study the well-posedness of a model problem of this type. Sufficient conditions are found to ensure global existence and finite time blowup. The blowup rate and the blowup set are also investigated in the case of special nonlinearity.

Global existence and non-existence problem for a reaction—diffusion equation with a conserved first integral

1995 ◽  
Vol 451 (1943) ◽  
pp. 701-709
Keyword(s):  
Global Existence ◽  
Diffusion Equation ◽  
Finite Time ◽  
Existence Of Solutions ◽  
Reaction Diffusion ◽  
First Integral ◽  
Stationary Solutions ◽  
Reaction Diffusion Equation ◽  
Existence Problem ◽  
Appl Math

In this paper, we prove the global existence and non-existence of solutions of the following problem: RDC{ u t = u xx + u 2 - ∫ u 2 ( x ) d x , x ϵ (0, 1), t > 0, u x (0, t ) = u x (1, t ) = t > 0, u ( x , 0) = u 0 ( x ), x ϵ (0, 1), ∫ 1 0 u ( x, t ) d x = 0, t > 0, Moreover, let u m ( x ) be a stationary solution of problem RDC with m zeros in the interval (0, 1) for m ϵ N , and if we take u 0 ( x ). Then we have that the solution exists globally if 0 < ϵ < 1, and blows up in finite time if ϵ > 1. This result verifies the numerical results of Budd et al . (1993, SIAM Jl appl. Math . 53, 718-742) that the non-zero stationary solutions are unstable.

A local theory for a fractional reaction-diffusion equation

2019 ◽  
Vol 21 (06) ◽  
pp. 1850033
Author(s):  
Arlúcio Viana
Keyword(s):  
Diffusion Equation ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Local Theory ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Singular Initial Data ◽  
Fractional Reaction

In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like [Formula: see text] and [Formula: see text], and [Formula: see text] is the characteristic function of a ball [Formula: see text]. Here, we are interested in the solvability of the problem when singular initial data [Formula: see text] are taken in [Formula: see text]. Eventually, we give sufficient conditions to the nonexistence of positive global solutions.

scholarly journals Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Dengming Liu ◽  
Chunlai Mu
Keyword(s):  
Boundary Condition ◽  
Global Existence ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Nonlocal Boundary Conditions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Subsolution And Supersolution

We investigate the blowup properties of the positive solutions for a semilinear reaction-diffusion system with nonlinear nonlocal boundary condition. We obtain some sufficient conditions for global existence and blowup by utilizing the method of subsolution and supersolution.

Global Existence and Blow-Up in Finite Time of Solutions to One Kind of Reaction-Diffusion Educations with Neumann Boundary Conditions

2014 ◽  
Vol 971-973 ◽  
pp. 1017-1020
Author(s):  
Jun Zhou Shao ◽  
Ji Jun Xu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Blowing Up ◽  
Processing Techniques

This paper deals with the properties of one kind of reaction-diffusion equations with Neumann boundary conditions based on the comparison principles. The relations of parameter and the situation of the coupled about equations are used to construct the global existent super-solutions and the blowing-up sub-solutions, and then we obtain the conditions of the global existence and blow-up in finite time solutions with the processing techniques of inequality.

scholarly journals Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chao Yang
Keyword(s):  
Global Existence ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Invariant Sets ◽  
Well Posedness ◽  
Sharp Condition ◽  
The Cauchy Problem ◽  
Inhomogeneous Nonlinear Schrödinger Equation ◽  
Time Blowup

<p style='text-indent:20px;'>This paper studies the Cauchy problem of Schrödinger equation with inhomogeneous nonlinear term <inline-formula><tex-math id="M1">\begin{document}$ V(x)|\varphi|^{p-1}\varphi $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. For the case <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1+\frac{4(1+\varepsilon_0)}{n} (0&lt;\varepsilon_0&lt;\frac{2}{n-2}) $\end{document}</tex-math></inline-formula>, by introducing a potential well, we obtain some invariant sets of solution and give a sharp condition of global existence and finite time blowup of solution; for the case <inline-formula><tex-math id="M4">\begin{document}$ p&lt;1+\frac{4}{n} $\end{document}</tex-math></inline-formula>, we obtain the global existence of solution for any initial data in <inline-formula><tex-math id="M5">\begin{document}$ H^1 (\mathbb{R}^n) $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Asymptotic behavior of the solutions of a discrete reaction-diffusion equation

2002 ◽  
Vol 29 (5) ◽  
pp. 257-264
Author(s):  
Rigoberto Medina ◽  
Sui Sun Cheng
Keyword(s):  
Asymptotic Behavior ◽  
Diffusion Equation ◽  
Asymptotic Representation ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Asymptotic Representation Of Solutions ◽  
Representation Of Solutions

By means of Bihari type inequalities, we derive sufficient conditions for solutions of a discrete reaction-diffusion equation to be bounded or to converge to zero. Asymptotic representation of solutions are also derived. Our results yield estimates and explicit attractive regions for the solutions.

scholarly journals On a generalized doubly parabolic Keller–Segel system in one spatial dimension

2015 ◽  
Vol 26 (01) ◽  
pp. 111-160 ◽  
Author(s):  
Jan Burczak ◽  
Rafael Granero-Belinchón
Keyword(s):  
Numerical Analysis ◽  
Upper Bound ◽  
Finite Time ◽  
Chaotic Behavior ◽  
Spatial Dimension ◽  
The Other ◽  
Well Posedness ◽  
Dynamical Properties ◽  
Spatio Temporal ◽  
Time Blowup

We study a doubly parabolic Keller–Segel system in one spatial dimension, with diffusions given by fractional Laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a numerical analysis suggesting that there is a finite time blowup if the diffusion is weak enough, even in presence of a damping logistic term. Our results generalize on one hand the results for local diffusions, on the other the results for the parabolic–elliptic fractional case.

Sign in / Sign up

Export Citation Format

Share Document