Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

On solutions to the Degasperis-Procesi equation

In this paper we consider a new integrable equation (the Degasperis-Procesi equation) derived recently by Degasperis and Procesi (1999) [3]. Analogous to the Camassa-Holm equation, this new equation admits blow-up phenomenon and infinite propagation speed. First, we give a proof for the blow-up criterion established by Zhou (2004) in [12]. Then, infinite propagation speed for the Degasperis-Procesi equation is proved in the following sense: the corresponding solution u(x,t) with compactly supported initial datum u₀(x) does not have compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t>0 in its lifespan, the corresponding solution u(x,t) behaves as: u(x,t)=L(t)e−x for x?1, and u(x,t)=l(t)ex for x?−1, with a strictly increasing function L(t)>0 and a strictly decreasing function l(t)<0 respectively.     

Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition

In this paper we consider a semilinear parabolic equation u_t=Δu−c(x,t)u^p for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣_{∂Ω×(0,∞)}=∫_Ωk(x,y,t)u^ldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.     

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.     

Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

This paper deals with u_t = Δu + u^m(x, t)e^pv(0,t), v_t = Δv + u^q(0, t)e^nv(x,t), subject to homogeneous Dirichlet boundary conditions. The complete classification on non-simultaneous and simultaneous blow-up is obtained by four sufficient and necessary conditions. It is interesting that, in some exponent region, large initial data u₀(v₀) leads to the blow-up of u(v), and in some betweenness, simultaneous blow-up occurs. For all of the nonnegative exponents, we find that u(v) blows up only at a single point if m > 1(n > 0), while u(v) blows up everywhere for 0 ? m ? 1 (n = 0). Moreover, blow-up rates are considered for both non-simultaneous and simultaneous blow-up solutions.     

On directional blow-up for quasilinear parabolic equations with fast diffusion

We discuss blow-up at space infinity of solutions to quasilinear parabolic equations of the form ut=Δ?(u)+f(u) with initial data u₀∈L^∞(RN), where ? and f are nonnegative functions satisfying ?^″?0 and . We study nonnegative blow-up solutions whose blow-up times coincide with those of solutions to the O.D.E. v^′=f(v) with initial data ‖u_0‖L^∞(RN). We prove that such a solution blows up only at space infinity and possesses blow-up directions and that they are completely characterized by behavior of initial data. Moreover, necessary and sufficient conditions on initial data for blow-up at minimal blow-up time are also investigated.     

Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u_t=Δu+u^p+u^q(x^*,t) in B(R) where p,q>0,B(R)={x∈R^N:|x|<R} and x^*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.

(i)

If or p=q>2, then single point blow-up occurs whenever solutions blow up.

(ii)

If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.

(iii)

If p?1<q, total blow-up occurs whenever solutions blow up.

(iv)

If max(p,q)?1, every solution exists globally in time.

     

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u?0 in Ω, with the boundary blow-up condition u_|∂Ω=+∞, where Ω is a bounded domain in RN(N?2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f₁) and (f₂) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.     

Blow-up phenomena for a semilinear heat equation with nonlinear boundary condition, I

We consider the blow-up of the solution to a semilinear heat equation with nonlinear boundary condition. We establish conditions on nonlinearities sufficient to guarantee that u(x, t) exists for all time t > 0 as well as conditions on data forcing the solution u(x, t) to blow up at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, lower bounds for t* are also derived.     

Boundary behaviour for solutions of boundary blow-up problems in a borderline case

We investigate boundary blow-up solutions of the equation Δu=f(u) in a bounded domain Ω⊂RN under the condition that f(t) has a relatively slow growth as t goes to infinity. We show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x) in terms of the distance of x from ∂Ω.     

Classification of type I and type II behaviors for a supercritical nonlinear heat equation

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut=Δu+|u|^p−1u either on RN or on a finite ball under the Dirichlet boundary conditions. We assume and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y,s) converges to either φ^∗(y) or −φ^∗(y) as s→∞, where φ^∗ denotes the singular stationary solution; (c) u(x,T)/φ^∗(x) tends to ±1 as x→0, where T is the blow-up time.Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L¹ continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, (t−T)^1/(p−1)‖u(⋅,t)L^∞_‖ is bounded as t↘T. We also give various criteria for complete and incomplete blow-ups.     

Existence and uniqueness of nonnegative solutions for a boundary blow-up problem

Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|^p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions.     

Similarity stabilizes blow up

The blow-up of solutions to the PDE ψ(x)u_t=[∇·A(x)∇+b(x)]u^m is studied via energy methods. The key step is a similarity transformation of the original unstable equation to a nonlocal stable one.     

Remarks on large solutions of a class of semilinear elliptic equations

In this paper, we show existence, uniqueness and exact asymptotic behavior of solutions near the boundary to a class of semilinear elliptic equations −Δu=λg(u)−b(x)f(u) in Ω, where λ is a real number, b(x)>0 in Ω and vanishes on ∂Ω. The special feature is to consider g(u) and f(u) to be regularly varying at infinity and b(x) is vanishing on the boundary with a more general rate function. The vanishing rate of b(x) determines the exact blow-up rate of the large solutions. And the exact blow-up rate allows us to obtain the uniqueness result.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

Bounds for blow-up time in nonlinear parabolic problems

A first order differential inequality technique is used on suitably defined auxiliary functions to determine lower bounds for blow-up time in initial-boundary value problems for parabolic equations of the form

ut=div(ρ(u)gradu)+f(u)     

Adaptive numerical schemes for a parabolic problem with blow-up

In this paper we present adaptive procedures for the numericalstudy of positive solutions of the following problem: u_t = u_xx (x, t) (0, 1) x [0, T), u_x(0, t) = 0 t [0, T), u_x(1, t) = u^p(1, t) t [0, T), u(x, 0) = u₀(x) x (0, 1), with p > 1. We describe two methods. The first one refinesthe mesh in the region where the solution becomes bigger ina precise way that allows us to recover the blow-up rate andthe blow-up set of the continuous problem. The second one combinesthe ideas used in the first one with moving mesh methods andmoves the last points when necessary. This scheme also recoversthe blow-up rate and set. Finally, we present numerical experimentsto illustrate the behaviour of both methods.     

Blow-up rates of radially symmetric large solutions

This paper adapts a technical device going back to [J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385-439] to ascertain the blow-up rate of the (unique) radially symmetric large solution given through the main theorem of [J. López-Gómez, Uniqueness of radially symmetric large solutions, Discrete Contin. Dyn. Syst., Supplement dedicated to the 6th AIMS Conference, Poitiers, France, 2007, pp. 677-686]. The requested underlying estimates are based upon the main theorem of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line, J. Differential Equations 244 (2008) 3180-3203]. Precisely, we show that if Ω is a ball, or an annulus, f∈C[0,∞) is positive and non-decreasing, V∈C[0,∞)∩C²(0,∞) satisfies V(0)=0, V^′(u)>0, V^″(u)?0, for every u>0, and V(u)∼Hup−1 as u↑∞, for some H>0 and p>1, then, for each λ?0,

−Δu=λu−f(dist(x,∂Ω))V(u)u     

Existence, non-existence and asymptotic behavior of blow-up entire solutions of semilinear elliptic equations

We are mainly concerned with existence, non-existence and the behavior at infinity of non-negative blow-up entire solutions of the equation Δu=ρ(x)f(u) in RN. No monotonicity condition is assumed upon f and, in fact, we obtain solutions with a prescribed behavior both at infinity and at the origin. The method used to get existence is based upon lower and upper solutions techniques while for non-existence we explore radial symmetry, estimates on an associated integral equation and the Keller-Osserman condition.     

Determination of the Unknown Cauchy Data in a Linear Parabolic Problem from the Measured Data at the Final Time

The problem of determining the initial value u(x, 0) = μ ₀(x) in the parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the final overdetermination μ _T (x) = u(x, T) is formulated. It is proved that the Fréchet derivative of the cost functional ${{J(\mu_0) = \|\mu_T(x) - u(x, T)\|_0^2}}$ can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. The existence of a quasisolution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method.