Complete blow-up of solutions for degenerate semilinear parabolic first initial-boundary value problems
Introduction
Let q be a nonnegative real number, D = (0, 1), Ωt = D × (0, t), , and Lu = xqut − uxx. We consider the following degenerate semilinear parabolic problem:where f and u0 are given functions.
A point x ∈ D is called a blow-up point of a function v(x, t) at tb if there exists a sequence (xm, tm) such thatIf v and t are interpreted, respectively, as the temperature and the time, then tb is the blow-up time. The physical interpretation of a blow-up is generally thought of as a dramatic increase in temperature which leads to an explosion or an ignition of a chemical reaction.
For q = 0, L is the heat operator; the problem (1.1), (1.2) (cf. [1, p. 10]) may be used to describe the temperature u(x, t) of a homogeneous and isotropic rod having a constant cross-sectional area with respect to x, and a thermal conductivity K independent of x; inside the rod, there is a nonlinear source producing heat (due to an exothermic reaction) at Kf(u) per unit volume per unit time; the object has an initial distribution of temperature u0(x), and the temperature at each of its ends is kept at zero. For q = 1, the problem (1.1), (1.2) may be used to describe the temperature u of the channel flow of a fluid with a temperature-dependent viscosity in the boundary layer (cf. [2], [3]); here, x and t denote the coordinates perpendicular and parallel to the channel wall, respectively; hence, tb corresponds to the downstream position where u blows up at some x. In a heat conduction problem with t denoting the time, the term xq corresponds to the reciprocal of the diffusivity (cf. [1, p. 9]); thus for q > 0, the amount of heat required to raise the temperature of the object approaches to zero as x tends to zero; also for a fixed x ∈ D, xq is a decreasing function of q; physically, decreasing x or increasing q has the effect of shifting the blow-up point towards x = 0.
For ease of reference, we interpret t as time in the sequel. We assume that f(0) ⩾ 0, f′(0) ⩾ 0, f″(u) > 0 for u > 0, and for some constant α ∈ (0, 1) is a nontrivial and nonnegative function such that in D, and u0(0) = 0 = u0(1). An argument similar to the proofs for Lemma 2 and Theorem 3 of Chan and Liu [4] gives the following result. Lemma 1.1 There exists some tb such that the problem (1.1), (1.2) has a unique solution . If tb < ∞, then u is unbounded in .
We note that under different conditions on f and u0, existence of a unique classical solution was proved by Chan and Chan [5] using a different method.
Green’s function G(x, t; ξ, τ) corresponding to the problem (1.1), (1.2) is determined by the following system: for x and ξ in D, and t and τ in (−∞, ∞),where δ(x) is the Dirac delta function. It was derived by Chan and Chan [5] thatwhere λi (i = 1, 2, 3, …) are the eigenvalues of the Sturm–Liouville problem,and their corresponding normalized eigenfunctions with the weight function xq are given bywith J1/(q+2) denoting the Bessel function of the first kind of order 1/(q + 2). From Chan and Chan [5], 0 < λ1 < λ2 < λ3 < ⋯ < λi < λi+1 < ⋯. The set {ϕi(x)} is a maximal (that is, complete) orthonormal set with the weight function xq (cf. [6, p. 176]).
For convenience of reference, let us state below the results of Lemmas 4 and 5 of Chan and Chan [5]. Lemma 1.2 (a) In {(x, t; ξ, τ) : x and ξ are in D, and t > τ ⩾ 0}, G(x, t; ξ, τ) is positive. (b) In , u ⩾ u0, and ut ⩾ 0.
To derive the integral equation from the problem (1.1), (1.2), let us consider the adjoint operator L*, which is given by L*u = −xqut − uxx. Using Green’s second identity, we obtain Lemma 1.3 Before a blow-up occurs, the integral equation (1.4) has at most one nonnegative solution. Proof Let Hu = ut − uxx, and be its corresponding Green’s function. Since ut ⩾ 0, we have Hu ⩾ Lu, and hence , which is positive by Lemma 1.2(a). On the other hand,(cf. [7, pp. 82–83]). Let U and V be two different solutions of the integral equation. Without loss of generality, let U > V somewhere, and η = min{t : U ⩾ V}. Let be denoted by ρ. From (1.4),Choosing t such that ρ(t − η) < 1, we have a contradiction. Thus, the lemma is proved. □
We say that U(x, t) is an integral solution (cf. [8]) of the problem (1.1), (1.2) if is a nonnegative measurable function satisfying (1.4) a.e. in . For t < tb < ∞, it follows from Lemma 1.1, Lemma 1.2(b) that the problem (1.1), (1.2) has a unique nonnegative solution u. Since a solution of the problem (1.1), (1.2) is a solution of the integral equation (1.4), it follows from Lemma 1.3 that for t < tb, u is the integral solution U. When t ⩾ tb, U(x, t) is a weak solution of the problem (1.1), (1.2); this includes the possibility that U is finite somewhere in D for t ⩾ tb.
If u can be continued after the blow-up time, then a true explosion or ignition does not take place (cf. [9]). To study a possible continuation after a blow-up, we consider a partial blow-up. The blow-up is called incomplete (cf. [9]) if there exists x ∈ D such that U(x, t) < ∞ for some t > tb. If no such x exists, then there is a complete blow-up. If u denotes the temperature of a chemical reaction described by the problem (1.1), (1.2), then the complete blow-up of U implies physically an ignition or a thermal explosion everywhere in D.
For a first initial-boundary value problem involving a n-dimensional semilinear heat equation, the complete blow-up was studied by Lacey and Tzanetis [10]. They proved that if the nonlinear forcing term f(u) is of the order o(u1+2/n) as u tends to infinity, and u blows up somewhere in a domain Dn at tb, then U(x, t) = ∞ in Dn for t > tb. Under different assumptions on f(u), the complete blow-up of U in Dn for t > tb was also given by Baras and Cohen [8]. More recently, Chan and Yang [11] studied the complete blow-up of the solution for the problem (1.1), (1.2) when the forcing term f is of the form f(u(x0, t)) for some x0 ∈ D; this describes the situation where the nonlinear reaction takes place only at the single point x0. They proved that the blow-up set is at tb. Our main purpose here is to study the complete blow-up of integral solutions for the degenerate semilinear parabolic problem (1.1), (1.2).
We note that in the special case f(u) = up, Floater [12] showed that if 1 < p ⩽ q + 1 andthen the solution of the problem (1.1), (1.2) blows up at x = 0. On the other hand, Chan and Liu [4] proved that if p > q + 1, and instead of (1.5), we have in D for some positive constant k, then x = 0 is not a blow-up point; furthermore, the blow-up set is a compact subset of D.
We show that any integral solution U of the problem (1.1), (1.2) blows up everywhere in D for t > tb. Our method is different from those of Baras and Cohen [8] and Lacey and Tzanetis [10], and unlike theirs, we allow the partial differential equation to be degenerate. Baras and Cohen assumed f(0) = 0 while Lacey and Tzanetis required f(0) > 0. Unlike theirs, we allow f(0) ⩾ 0; this allows, for example, f(u) to include, as a special case, the commonly studied power function up in the blow-up of solutions.
Section snippets
Complete blow-up
Let λ be the fundamental eigenvalue of the Sturm–Liouville problem (1.3), and ϕ(x) denote its corresponding normalized eigenfunction with the weight function xq. By Chan and Chan [5], ϕ(x) is positive in D. It follows from f(0) ⩾ 0, f′(0) ⩾ 0, and f″(u) > 0 for u > 0 that f(u) > 0 for u > 0. Let R(s) = f(s)/2 − λs, r denote the largest positive root of R(s) = 0 if it exists, andAlso, let . We modify the proof of Theorem 8 of Kaplan [13] to obtain the following
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