Positive solutions of reaction diffusion equations with super-linear absorption: Universal bounds, uniqueness for the Cauchy problem, boundedness of stationary solutions

Consider classical solutions u∈C²(Rⁿ×(0,∞))∩C(Rⁿ×[0,∞)) to the parabolic reaction diffusion equation

     

A nonlinear parabolic equation with a nonlocal boundary term

A nonlinear parabolic problem with a nonlocal boundary condition is studied. We prove the existence of a solution for a monotonically increasing and Lipschitz continuous nonlinearity. The approximation method is based on Rothe’s method. The solution on each time step is obtained by iterations, convergence of which is shown using a fixed-point argument. The space discretization relies on FEM. Theoretical results are supported by numerical experiments.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

On a reaction diffusion equation with nonlinear time‐nonlocal source term

In this article, we prove the local existence of a unique solution to a nonlocal in time and space evolution equation with a time nonlocal nonlinearity of exponential growth. Moreover, under some suitable conditions on the initial data, it is shown that local solutions experience blow‐up. The time profile of the blowing‐up solutions is also presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

Boundary fluxes for nonlocal diffusion

We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition.     

Growup of solutions for a semilinear heat equation with supercritical nonlinearity

We consider a Cauchy problem for a semilinear heat equation

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Boundedness of global solutions of a porous medium equation with a localized source

In this paper a localized porous medium equation _ut=u^r(Δu+af(u(_x0,t)))$u_t=u^r(\mathrm{\Delta }u+\mathit{af}(u(x_0,t)))$ is considered. It is shown that under certain conditions solutions of the above equation blow up in finite time for large a or large initial data while there exist global positive solutions to the above equation for small a or small initial data. Moreover, it is also shown that all global positive solutions of the above equation are uniformly bounded, and this differs from that of a porous medium equation with a local source.     

Multiple blowup of solutions for a semilinear heat equation II

The present paper is concerned with a Cauchy problem for a semilinear heat equation

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Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition

Let p>1 and Ω be a smoothly bounded domain in . This paper is concerned with a Cauchy-Neumann problem

     

Various behaviors of solutions for a semilinear heat equation after blowup

The present paper is concerned with a Cauchy problem for a semilinear heat equation

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Weak solutions for nonlocal boundary value problems with low regularity data

This paper concerns the existence and uniqueness of weak solutions for elliptic and parabolic equations under nonlocal boundary conditions, based on maximal regularity. It also gives the positivity of solutions which can be used in monotone iteration methods. As an application, the results are used to discuss some specific nonlocal problems.     

Regular traveling waves for a nonlocal diffusion equation

In this paper, we study a nonlocal diffusion equation with a general diffusion kernel and delayed nonlinearity, and obtain the existence, nonexistence and uniqueness of the regular traveling wave solutions for this nonlocal diffusion equation. As an application of the results, we reconsider some models arising from population dynamics, epidemiology and neural network. It is shown that there exist regular traveling wave solutions for these models, respectively. This generalized and improved some results in literatures.     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Numerical solution of a Cauchy problem for nonlinear reaction diffusion processes

In this paper, the authors propose a numerical method to compute the solution of the Cauchy problem: _wt-(w^m_wx)x=w^p$w_t-(w^m{w}_x{)}_x=w^p$, the initial condition is a nonnegative function with compact support, m>0$m>0$, p?m+1$p?m+1$. The problem is split into two parts: a hyperbolic term solved by using the Hopf and Lax formula and a parabolic term solved by a backward linearized Euler method in time and a finite element method in space. The convergence of the scheme is obtained. Further, it is proved that if m+1?p<m+3$m+1?p<m+3$, any numerical solution blows up in a finite time as the exact solution, while for p>m+3$p>m+3$, if the initial condition is sufficiently small, a global numerical solution exists, and if p?m+3$p?m+3$, for large initial condition, the solution is unbounded.     

The instability of spiky steady states for a competing species model with cross diffusion

This paper is concerned with the stability/instability of a class of positive spiky steady states for a quasi-linear cross-diffusion system describing two-species competition. By detailed spectral analysis, it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross-diffusion system are non-linearly unstable.     

Existence and uniqueness of solutions for a non-uniformly parabolic equation

In this paper, we study a generalized Burgers equation u_t+(u²)_x=tu_xx, which is a non-uniformly parabolic equation for t>0. We show the existence and uniqueness of classical solutions to the initial-value problem of the generalized Burgers equation with rough initial data belonging to .     

Critical exponents and blow-up rate for a nonlinear diffusion equation with logarithmic boundary flux

In this paper, we establish the critical global existence exponent and the critical Fujita exponent for the nonlinear diffusion equation u_t=(log^σ(1+u)u_x)_x, in R₊×(0,+∞), subject to a logarithmic boundary flux , furthermore give the blow-up rate for the nonglobal solutions.