Thermal avalanche for blowup solutions of semilinear heat equations

We consider the semilinear heat equation u_t = Δu + u^p both in ?^N and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < p_s where p_s is the Sobolev exponent. This problem has solutions with finite‐time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well‐known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T^? and t = T⁺. This is called the “avalanche.” We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power u^p by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.     

The finite-time blowup of the solution of an initial boundary-value problem for the nonlinear equation of ion sound waves

We obtain blowup conditions for the solutions of initial boundary-value problems for the nonlinear equation of ion sound waves in a hydrogen plasma in the approximation of “hot” electrons and “heavy” ions. A specific characteristic of this nonlinear equation is the noncoercive nonlinearity of the form ?_t|?u|², which complicates its study by any energy method. We solve this problem by the Mitidieri–Pohozaev method of nonlinear capacity.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

A sufficient condition for blowup solutions of nonlinear heat equations

The author discusses the initial-boundary value problem (u_i)_t=Δu_i+f_i(u₁,…,u_m) with and u_i(x,0)=φ_i(x), i=1,…,m, in a bounded domain Ω⊂Rn. Under suitable assumptions on f_i, he proves that, if φ_i?(1+ε₀)ψ_i in , for some small ε₀>0, then the solutions blow up in a finite time, where ψ_i is a positive solution of Δψ_i+f_i(ψ₁,…,ψ_m)?0, with ψ_i|_∂Di=0 for i=1,…,m. If m=1, the initial value can be negative in a subset of Ω.     

Analytical blowup solutions to the 2-dimensional isothermal Euler-Poisson equations of gaseous stars

We study the Euler-Poisson equations of describing the evolution of the gaseous star in astrophysics. Firstly, we construct a family of analytical blowup solutions for the isothermal case in R². Furthermore the blowup rate of the above solutions is also studied and some remarks about the applicability of such solutions to the Navier-Stokes-Poisson equations and the drift-diffusion model in semiconductors are included. Finally, for the isothermal case (γ=1), the result of Makino and Perthame for the tame solutions is extended to show that the life span of such solutions must be finite if the initial data is with compact support.     

On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory

Theoretical and Mathematical Physics - We study two Cauchy problems for nonlinear equations of the Sobolev type, of the form $$ frac{partial}{partial t}frac{partial^2u}{partial x_3^2} +...     

On constant-sign solutions of a system of discrete equations

We consider the following system of discrete equations $$u_i (k) = \sum\limits_{\ell = 0}^N {g_i (k,\ell )fi(\ell ,u_1 (\ell )} ,u_2 (\ell ), \cdots ,u_n (\ell )), k \in \{ 0,1, \cdots ,T\} ,$$ 1≤i≤n whereT≥N>0, 1≤i≤n. Existence criteria for single, double and multiple constant-sign solutions of the system are established. To illustrate the generality of the results obtained, we include applications to several well known boundary value problems. The above system is also extended to that on {0, 1,…} $$u_i (k) = \sum\limits_{\ell = 0}^\infty {g_i (k,\ell )fi(\ell ,u_1 (\ell )} ,u_2 (\ell ), \cdots ,u_n (\ell )), k \in \{ 0,1, \cdots \} ,1 \leqslant i \leqslant n$$ for which the existence of constant-sign solutions is investigated.     

Continuation of blowup solutions of nonlinear heat equations in several space dimensions

The possible continuation of solutions of the nonlinear heat equation in R^N × R₊ u_t = Δu^m + u^p with m > 0, p > 1, after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p. Thus, for m + p ≤ 2 we find a phenomenon of nontrivial continuation where the region {x : u(x, t) = ∞} is bounded and propagates with finite speed. This we call incomplete blowup. For N ≥ 3 and p > m(N + 2)/(N − 2) we find solutions that blow up at finite t = T and then become bounded again for t > T. Otherwise, we find that blowup is complete for a wide class of initial data. In the analysis of the behavior for large p, a list of critical exponents appears whose role is described. We also discuss a number of related problems and equations. We apply the same technique of analysis to the problem of continuation after the onset of extinction, for example, for the equation u_t = Δu^m − u^p, m > 0. We find that no continuation exists if p + m ≤ 0 (complete extinction), and there exists a nontrivial continuation if p + m > 0 (incomplete extinction). © 1997 John Wiley & Sons, Inc.     

Global existence and blowup of solutions for a class of nonlinear higher-order wave equations

In this paper, we consider a class of nonlinear higher-order wave equation with nonlinear damping $$u_{tt}+(-\Delta)^mu+a|u_t|^{p-2}u_t=b|u|^{q-2}u$$ in a bounded domain ${\Omega\subset\mathbb{R}^N}$ (N????1 is a natural number). We show that the solution is global in time under some conditions without the relation between p and q and we also show that the local solution blows up in finite time if q?>?p with some assumptions on initial energy. The decay estimate of the energy function for the global solution and the lifespan for the blow-up solution are given. This extend the recent results of Ye (J Ineq Appl, 2010).     

On the regularity of solutions to a parabolic system related to Maxwell's equations

The goal of this paper is to establish Hölder estimates for the solutions of a certain parabolic system related to Maxwell's equations arising in a quasi-stationary electromagnetic field.     

Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation in (0.1) $ u_t = (uu_x )_{xx} in\mathbb{R} \times \mathbb{R}_ + . $ It is shown that two basic Riemann problems for Eq. (0.1) with the initial data $ S_ \mp (x) = \mp \operatorname{sgn} x $ exhibit a shock wave (u(x, t) ≡ S _?(x)) and a smooth rarefaction wave (for S ₊), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u _t + uu _x = 0, which was developed by O.A. Oleinik and S.N. Kruzhkov (for equations in x ? ? ^N ) in the 1950s–1960s.     

On the solutions of a system of two Diophantine equations

We obtain all positive integer solutions(m1,m2,a,b)with ab,gcd(a,b)=1 to the system of Diophantine equations km21-lat1bt2a2r=C1,km22-lat1bt2b2r=C2,with C1,C2∈{-1,1,-2,2,-4,4},and k,l,t1,t2,r∈Z such that k0,l0,r0,t10,t2 0,gcd(k,l)=1,and k is square-free.     

A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term

A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term is established in this paper. In virtue of the potential wells method, we first extend the results obtained by Xu and Su in [J. Funct. Anal., 264 (12): 2732‐2763, 2013] to the nonlocal case and describe successfully the behavior of solutions by using the energy functional, Nehari functional, and the ground state energy of the stationary equation. Sequently, we study the boundedness and convergency of any global solution. Finally, we achieve a criterion to guarantee the blowup of solutions without any limit of the initial energy.Copyright © 2015 John Wiley & Sons, Ltd.