Blow-up properties for a degenerate parabolic system with nonlinear localized sources

In this paper, we investigate the initial-boundary problem of a degenerate parabolic system with nonlinear localized sources. We classify the blow-up solutions into global blow-up cases and single-point blow-up cases according to the values of m,n,p_i,q_i. Furthermore, we obtain the uniform blow-up profiles of solutions for the global blow-up case. Finally, we give some numerical examples to verify the results. These extend and generalize a recent work of one of the authors [L. Du, Blow-up for a degenerate reaction-diffusion systems with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320], which only considered uniform blow-up profiles under the special case p₁=p₂=0.     

Blow-up properties of solutions for a multi-coupled parabolic system

This paper deals with the blow-up behavior of radial solutions to a parabolic system multi-coupled via inner sources and boundary flux. We first obtain a necessary and sufficient condition for the existence of non-simultaneous blow-up, and then find five regions of exponent parameters where both non-simultaneous and simultaneous blow-up may happen. In particular, nine simultaneous blow-up rates are established for different regions of parameters. It is interesting to observe that different initial data may lead to different simultaneous blow-up rates even with the same exponent parameters.     

Existence for a degenerate diffusion problem with a nonlinear operator

We study a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function. We establish existence, uniqueness and regularity results. An application arising in the theory of water diffusion in porous media is discussed as an example.      

Blow-up sets for linear diffusion equations in one dimension

We consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval [0,a], depending on the Dirichlet data, we prove that the effective blow-up set, that is, the set of points where the solution behaves like u(0,t), consists always only of the origin. As an application of our results we consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. We show that by prescribing the non-linearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.     

Blow-up of solutions to a degenerate parabolic equation not in divergence form

We study nonglobal positive solutions to the Dirichlet problem for u_t=u^p(Δu+u) in bounded domains, where 0<p<2. It is proved that the set of points at which u blows up has positive measure and the blow-up rate is exactly . If either the space dimension is one or p<1, the ω-limit set of consists of continuous functions solving . In one space dimension it is shown that actually as t→T, where w coincides with an element of a one-parameter family of functions inside each component of its positivity set; furthermore, we study the size of the components of {w>0} with the result that this size is uniquely determined by Ω in the case p<1, while for p>1, the positivity set can have the maximum possible size for certain initial data, but it may also be arbitrarily close to the minimal length π.     

On nonlinear diffusion problems with strong degeneracy

In this paper, we study the “triply” degenerate problem: bt(v)−Δg(v)+divΦ(v)=f on Q:=(0,T)×Ω, b(v(0,⋅))=b(v₀) on Ω and “g(v)=g(a) on some part of the boundary (0,T)×∂Ω,” in the case of continuous nonhomogeneous and nonstationary boundary data a. The functions b,g are assumed to be continuous, locally Lipschitz, nondecreasing and to verify the normalization condition b(0)=g(0)=0 and the range condition R(b+g)=R. Using monotonicity and penalization methods, we prove existence of a weak renormalized entropy solution in the spirit of [K. Ammar, J. Carrillo, P. Wittbold, Scalar conservation laws with general boundary condition and continuous flux function, J. Differential Equations 228 (2006) 111-139].     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

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.     

Stefan problems with nonlinear diffusion and convection

We consider a class of Stefan-type problems having a convection term and a pseudomonotone nonlinear diffusion operator. Assuming data in L¹, we prove existence, uniqueness and stability in the framework of renormalized solutions. Existence is established from compactness and monotonicity arguments which yield stability of solutions with respect to L¹ convergence of the data. Uniqueness is proved through a classical L¹-contraction principle, obtained by a refinement of the doubling variable technique which allows us to extend previous results to a more general class of nonlinear possibly degenerate operators.     

Blow-up and life span estimates for a class of nonlinear degenerate parabolic system with time-dependent coefficients

This paper deals with the singularity and global regularity for a class of nonlinear porous medium system with time-dependent coefficients under homogeneous Dirichlet boundary conditions. First, by comparison principle, some global regularity results are established. Secondly, using some differential inequality technique, we investigate the blow-up solution to the initial-boundary value problem. Furthermore, upper and lower bounds for the maximum blow-up time under some appropriate hypotheses are derived as long as blow-up occurs.     

Existence of solutions of the very fast diffusion equation in bounded and unbounded domain

We prove the existence of a unique solution of the following Neumann problem , u > 0, in (a, b) × (0, T), u(x, 0) = u ₀(x) ≥ 0 in (a, b), and , where if m < 0, if m = 0, and m≤ 0, , and the case −1 < m ≤ 0, , for some constant p > 1 − m. We also obtain a similar result in higher dimensions. As a corollary we will give a new proof of a result of A. Rodriguez and J.L. Vazquez on the existence of infinitely many finite mass solutions of the above equation in for any −1 < m ≤ 0. We also obtain the exact decay rate of the solution at infinity.     

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.     

Approaching a vertex in a shrinking domain under a nonlinear flow

We consider here the homogeneous Dirichlet problem for the equation , in a noncylindrical domain in space-time given by . By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x=0, t=T, in the three different cases p>1/2, p=1/2(vertex regular), p<1/2 (vertex irregular).     

Blow-up phenomena for some nonlinear parabolic problems

We consider the blow-up of solutions of equations of the form

_ut=div(ρ(|∇u|²) grad u)+f(u)$u_t=\text{div}\left(\rho \left({\left|\nabla u\right|}^2\right)\text{ grad }u\right)+f\left(u\right)$

Blow-up analysis for a nonlinear diffusion system

This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.     

Blow-up phenomena for a doubly degenerate equation with positive initial energy

In this paper, an initial boundary value problem related to the equation

     

Convergence to steady states in a viscous Hamilton-Jacobi equation with degenerate diffusion

This paper deals with weak solutions of the one-dimensional viscous Hamilton-Jacobi equation

     

Blow-up for semilinear parabolic equations with nonlinear memory

In this paper, we consider the semilinear parabolic equation with homogeneous Dirichlet boundary conditions, where p, q are nonnegative constants. The blowup criteria and the blowup rate are obtained.     

The critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux

This paper deals with the critical curve of the non-Newtonian polytropic filtration equation coupled via nonlinear boundary flux. The critical global existence curve is obtained by constructing various self-similar supersolutions and subsolutions. Furthermore, we get some new results on the critical Fujita curve.