| In this paper we study the following problem: where p>2; = (0,l)isa finite interval when the space dimension N = 1, and RN is a ball of radius l with N > 2; n is the outward unit normal vector on the boundary of f and g are smooth functions which are positive when the argument is positive; and u0(x) > 0 satisfies some smooth and compatibility conditions. We consider classical solutions only. Thus, when we say u is a solution, we always assume u C2,1(Qt) C1,0(QT). In addition, we only consider positive solutions in this entire paper. A positive solution of parabolic or elliptic problems means that the solution is strictly positive in QT or in respectively.This paper cosists five parts: the first part provides us the toolof this paper - the comprison principle; and the remaining parts prove the main results of this paper.(/) The comprison principle.First of all, we have the comprison principle to the following problem:LEMMA 1 If u is a solution of (1) -(3), and u is a e-subsolutionof (1) - (3) with u(x,0) < u0(x), then u(x,t) < u(x,t) holds for all x [0, l] and t (0, Tmax), where Tmax is the maximun time that u exists.Then the radial solutions of the problem (1) - (3) can be written as the solutions of the following problem:can also be proven the comprison principle which satisfies the above problem.LEMMA 2 If u is a solution of (7) - (9), and u is a e-subsolution of (7) - (9) with u(x, 0) < u0(x), then u(x, t) < u(x, t) holds for all x [0,l] and t (0,Tmax), where Tmax is the maximum time thatu exists.The main results of this paper are the followings.(II) blow-up at a finite time.At first, we obtain the existence of the solution to the following problem:Then, we prove that the theorem of blow-up at a finite time in the case N= 1.THEOREM 1 Let u(x, t) C2,1(QT) C1,0(QT) be a positive solution of (4) - (6), Suppose that there exists a continously differentiable function m(u) for 0 < u < such that m(0) -m0, 0 < m0 < 1, m'{u) >0, andfor some positive constant >1. Then the solution will blow up in finite time if u0 > 0 is large enough, for the above constant To prove the theorem, we construct a special e-subsolution u(x,t) = v(t + h(x)) firstly with v and h satisfies some differential equations and conditions, By computation and the existence of thesolution to problem (10) - (11) we know that u is a e-subsolution solution to problem (4) - (6) for large u0(x). Since we may prove that the e-subsolution v(s) blows up in finite time, we conclude from standard comprison theory that u(x, t) blows up in finite time. This completes the proof.In the case N > 2 we see that the radial solution satisfies the problem (7) - (9) a direct calculation. So we obtain the following theorem by a similar proof of the above theorem.THEOREM 2 Let u(x, t) C2,1(QT) C1,0(QT) be a positive solution of (4) - (6). Suppose that there exists a continously differentiable function m(u) for 0 < u < oo such that m(0) = m0, 0 < m0 < 1, m'(u) >0, andfor some positive constant . Then the solution will blow up in finite time if u0 > 0 is large enough, for the above constant and (III) The result of global finiteness.Under the assumption that there exists a sufficiently small constant > 0 such that f(u) > 2 > 0 and set F(u) be the integral , we may prove the global finiteness for the solution in the case N = 1.THEOREM 3 Assume that for any fixed constant a, b > 0, there is a constant A such that if t > A, thenMoreover, for any the following inequality holds,Then every positive solution of (4) - (6) is finite for (x,t) [0, ).Our idea is to find an arbitrarily large e-supersolution for problem (4) - (6) to prove that the solution is finite for all the time. By computation and using the result of the existence of the solution to problem (12) - (13) we can construct any arbitrarily large e-supersolution. We conclude that every solution is finite for all the time.In the case N > 2 we see that the radial solution satisfies the problem (7) - (9) by a direct calc... |
PDF Full Text Request