EXISTENCE, UNIQUENESS AND QUENCHING OF THE SOLUTION FOR A NONLOCAL DEGENERATE SEMILINEAR PARABOLIC PROBLEM

Transcription

1 Dynamic Systems and Applications 6 (7) EXISTENCE, UNIQUENESS AND QUENCHING OF THE SOLUTION FOR A NONLOCAL DEGENERATE SEMILINEAR PARABOLIC PROBLEM C. Y. CHAN AND H. T. LIU Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 754-, USA (chan@louisiana.edu) Department of Applied Mathematics, Tatung University, 4 Chung Shan North Road, Sec. 3, Taipei, Taiwan 4 (tliu@ttu.edu.tw) ABSTRACT. Let a and T be positive constants, D = (, a), D = [, a], Ω = D (, T ], and Lu = x q u t u xx, where q is a nonnegative number. This article studies the following problem, Lu(x, t) = k(y)f(u(y, t))dy in Ω, where k is a positive function on D, f >, f, f, and lim u f(u) =, subject to the initial condition u(x, ) = on D, and the boundary conditions u(, t) = = u(a, t) for < t T. Existence of a unique solution, the critical length, and the quenching behavior of the solution are studied. AMS (MOS) Subject Classification. 35K57, 35K6, 35K65.. INTRODUCTION Let a and T be constants, D = (, a), D = [, a], Ω = D (, T ], Ω be the parabolic boundary, and Lu = x q u t u xx, where q is a nonnegative number. We consider the following nonlocal initial-boundary value problem, (.) Lu(x, t) = k(y)f(u(y, t))dy in Ω, (.) u(x, ) = on D, u(, t) = = u(a, ) for < t T, where k is a positive function on D, f >, f >, f, and lim u f(u) =. Chan and Kong [], and Chan and Liu [3] studied existence, uniqueness and quenching behavior of the solution u in the case k(y)f(u(y, t))dy being replaced by f(u). We show that the problem (.)-(.) has a unique classical solution, and give a criterion for quenching to occur and for global existence. Received January 8, $5. c Dynamic Publishers, Inc.

2 55 C. Y. CHAN AND H. T. LIU. EXISTENCE AND UNIQUENESS Since k(x) > on D, we have k(y)f(u(y, t))dy > for x >. From the strong maximum principle (cf. Friedman [5]), u > in Ω. We now prove the comparison results. Lemma.. Let w be a function such that Lw > g(y, t)w(y, t)dy in Ω, where g(x, t) is a bounded nonnegative function on Ω, and w > on Ω, then w > on Ω. Proof. Suppose that w somewhere on Ω. Let t = inf{t : w(x, t) for some x D}. Since w > on Ω, we have t >, and there exists some x D such that w( x, t) =, w(x, t) on D, w t ( x, t), and w xx ( x, t). This implies x q w t ( x, t) > w xx ( x, t) + x g(y, t)w(y, t)dy. We have a contradiction. Thus, w > on Ω. Theorem.. If w satisfies the inequality Lw g(y, t)w(y, t)dy in Ω, where g(x, t) is a bounded nonnegative function on Ω, and w on Ω, then w on Ω. Proof. For a fixed positive number η, let V (x, t) = w(x, t) + η( + x )e ct, where c is some positive constant to be determined. Since g(x, t) is bounded on Ω, let M = t [,T ] { g(x, t)( + x/ )dx }. We have V xx = w xx ηx 3/ e ct /4. Let s be the first positive zero of x 3/ /4 M. If s a, then for any x D, (.) cx q ( + x ) x M >. If s < a, then for any x (, s), the inequality (.) holds. For x [s, a], let us choose c such that c > M/s q. Then on [s, a], cx q (+x / ) > M(+x / ) > M > M x 3/ /4, and the inequality (.) holds. Thus for any x D, x q cη( + x )e ct > ηe ( ct 4 x 3 + M ) > 4 ηx 3 e ct + ηe ct g(x, t)( + x )dx 4 ηx 3 e ct + ηe ct g(y, t)( + y )dy.

3 This gives x q V t > w xx + EXISTENCE, UNIQUENESS AND QUENCHING 553 g(y, t)w(y, t)dy 4 ηx 3 e ct + ηe ct g(y, t)( + y )dy = w xx 4 ηx 3 e ct + = V xx (x, t) + [ ] g(y, t) w(y, t) + η( + y )e ct dy g(y, t)v (y, t) dy. Since w on Ω, we have V > on Ω. By Lemma., we have V > on Ω. As η, we obtain w on Ω. Theorem.3. Suppose that u is a solution of the problem (.)-(.), and v satisfies then v u on Ω. Lv k(y)f(v(y, t))dy in Ω, v on Ω, Proof. We have L(v u) k(y)(f(v) f(u))dy. By the mean value theorem, L(v u) k(y)(f (ξ)(v(y, t) u(y, t))dy for some ξ between u and v. By Theorem., v u on Ω. As a consequence of the comparison theorem, we have the following results. Theorem.4. The problem (.)-(.) has at most one solution. Let u a denote the solution of the problem (.)-(.). Theorem.5. If a > a, then u a (x, t) u a (x, t) for (x, t) [, a ] [, T ]. Proof. We have u a (, t) = and u a (a, t). Since u a (, t) = = u a (a, t), it follows from Theorem.3 that u a (x, t) u a (x, t). closure. We now show existence of the solution. Let Ω t = D (, t ], and Ω t be its Theorem.6. There exists some t (> ) such that the problem (.)-(.) has a unique nonnegative solution u C( Ω t ) C, (Ω t ). Proof. Let and t be positive constants with < a, Ω = (, a) (, t ], S = {, a} (, t ], Ω be the closures of Ω, and u be the solution of the problem, Lu = k(y)f(u (y, t))dy in Ω, (.) u (x, ) = on [, a]; u (x, t) = on S.

4 554 C. Y. CHAN AND H. T. LIU Let us construct an upper solution h(x, t) for all u as follows: (i) Let θ(x) = x γ (a x) γ, where γ (, ); also let k be a positive constant such that > k θ(x). (ii) Let ɛ be a positive number such that k θ(x) < ɛ <. (iii) Since θ (x) tends to as x tends to or a, there exists a positive number k D such that k θ (x) + f( ɛ) k(y)dy for x (, k ) (a k, a). (iv) Let g(t) be the solution of the initial value problem, ( kg q (t)θ(k ) = k(y)dy ) f ( (a ) γ g(t) ), g() = k. (v) Since (a/) γ k < ɛ and g (t) >, we can choose t > such that (a/) γ g(t ) = ɛ. To show that h(x, t) = θ(x)g(t) is an upper solution of u, let J = Lh By a direct computation, J(x, t) = x q θ(x)g (t) θ (x)g(t) For x (, k ) and t (, t ], J(x, t) θ (x)g(t) k θ (x) f( ɛ). For x [k, a k ] and t (, t ], k(y)f (θ(y)g(t)) dy. k(y)f(θ(y)g(t))dy k(y)dy J(x, t) x q θ(x)g (t) k(y)f(θ(y)g(t))dy ( ) ( (a ) ) γ kθ(k q )g (t) k(y)dy f g(t) =. For x (a k, a) and t (, t ], J(x, t) θ (x)g(t) k θ (x) f( ɛ). k(y)f(θ(y)g(t))dy k(y)dy k(y)f(h(y, t))dy. Now, h(x, ) = k θ(x). Since h(, t) >, and h(a, t) =, it follows from Theorem.3 that h is an upper solution.

5 EXISTENCE, UNIQUENESS AND QUENCHING 555 We note that x C α,α/ ( Ω ), x k(y)f(u (y, t))dy k(y)f(u (y, t))dy for (x, t, u) Ω R. Since v is a lower solution, it follows from Theorem 4.. of Ladde, Lakshmikantham and Vatsala [6, p.43] that the problem (.) has a unique solution u C +α,+α/ ( Ω ). Since u < u in Ω for >, lim u exists for all (x, t) Ω t. For any (x, t ) Ω t, there is a set Q = [b, b ] [t, t 3 ] Ω t, where b, b, t and t 3 are positive numbers such that b < x < b < a and t < t t 3. Since u h(x, t) in Q and h(x, t) <, we have for some constant p >, and some positive constants k 3 and k 4, u L p (Q) h(x, t) L p (Q) k 3, x k(y)f(u (y, t))dy Lp (Q) b k(y)f(h(y, t))dy Lp (Q) k 4. By Ladyženskaja, Solonnikov and Ural ceva [7, pp ], u Wp, (Q). By the embedding theorems there [7, pp. 6 and 8], Wp, (Q) H α,α/ (Q) by choosing p > /( α) with α (, ). Then, u H α,α/ (Q) k 5 for some constant k 5. For x < x, x k(y)f(u (y, t))dy H α,α/ (Q) b k(y)f(h(y, t))dy + x k(y)f(u (y, t))dy x k(y)f(u (y, t))dy x x α + (x, t ) Q (x, t ) Q x k(y)f(u (y, t ))dy k(y)f(u (y, t ))dy t t α/, the first term of which is bounded while the second term, x k(y)f(u (y, t))dy x k(y)f(u (y, t))dy x x α x k(y)f(u (y, t))dy x k(y)f(u (y, t))dy x x α

6 556 C. Y. CHAN AND H. T. LIU x + + k(x) D k(y)f(u (y, t))dy x x x α k(y)f(u (y, t))dy x x x x α x D [,t ] x k(y)f(u (y, t))dy x x α f(u (x, t)) k(y)f(u (y, t))dy x x x x x α + b k(x) D D [,t ] a k(x) D D [,t ] f(u (x, t)) f(h (x, t)) x H α,α/ (Q) x x x x α + b k(x) D D [,t ] f(h(x, t)) x x α k 6 for some constant k 6, and the last term, (x, t ) Q (x, t ) Q b x k(y)f(u (y, t ))dy k(y)f(u (y, t ))dy t t α/ k(y)f (h(y, t))dy (x, t ) Q k 7 for some constant k 7. (x, t ) Q u (x, t ) u (x, t ) t t α/

7 EXISTENCE, UNIQUENESS AND QUENCHING 557 Hence, x k(y)f(u (y, t))dy H α,α/ (Q) k 8 for some constant k 8 which is independent of. By Theorem 4.. of Ladyženskaja, Solonnikov and Ural ceva [7, pp ], we have u H +α,+α/ (Q) K for some constant K, which is independent of. This implies that u, (u ) t, (u ) x and (u ) xx are equicontinuous in Q. By the Ascoli-Arzela theorem, u H +α,+α/ (Q) K, and the partial derivatives of u are the limits of the corresponding partial derivatives of u. Thus, u C( Ω t ) C, (Ω t ). Let T = { t : the problem (.)-(.) has a solution on D [, t)}. A proof similar to that of Theorem.5 of Floater [4] gives the following result. Theorem.7. There is a unique solution u C( D [, T )) C, (D (, T )). If T <, then u tends to as t T. Existence of an upper solution guarantees a global existence result of the solution. Theorem.8. For a sufficiently small, the solution u exists globally. Proof. Let w(x) = ε(a x ), where ε is a positive number such that εa /4 <. Then w xx + k(y)f(w(y))dy = ε + k(y)f(w(y))dy ( ) εa ε + f k(y)dy. 4 Since k(y)dy as a, the right-hand side of the above inequality is negative when a is small. On the other hand, w() >, and w(a) =. This implies w is an upper solution which is bounded away from when a is small. 3. QUENCHING Let us consider the Sturm-Liouville problem, ϕ + λx q ϕ =, ϕ() = = ϕ(a). For q =, the eigenfunctions exist. For q >, Chan and Chan [] showed that the eigenfunctions are given by ( ) ( ) φ i (z) = / z /(q+) λ / i J /(q+) q + z J λ / i [/(q+)]+ q +, where z = x (q+)/, i =,, 3,, and J /(q+) is the Bessel function of the first kind of order /(q + ). Since { φ i (z)} forms an orthonormal set with the weight function z q/(q+), we have {φ i (x)} forms an orthonormal set with the weight function x q. Let

8 558 C. Y. CHAN AND H. T. LIU β denote the first positive zero of J /(q+), ϕ(x) be the fundamental eigenfunction with xq ϕ(x)dx =, and µ be the fundamental eigenvalue. By ϕ(a) =, we have (µ / a (q+)/ )/(q + ) = β. This gives µa q = a 3 [β(q + )/]. Hence, µa q decreases when a increases. We now give a condition for the solution u to quench in a finite time. Theorem 3.. If f() inf k(x) > µa q, then the solution u of the problem (.)-(.) quenches in a finite time. Proof. Let F (t) = xq ϕ(x)u(x, t)dx. Then, ( ) F (t) = u xx (x, t) + k (y) f (u(y, t)) dy ϕ(x)dx = u(x, t)ϕ (x)dx + ϕ(x) µf (t) + f() inf k(x) ϕ(x)xdx k (y) f (u(y, t)) dydx By a direct calculation, µf (t) + f() inf k(x). aq inf k(x) F (t) f() µa ( q e µt ). Since f() inf k(x) > µa q, there exists t > such that F (t ). Hence, u quenches somewhere in a finite time. We note that the condition f() inf k(x) > µa q can be satisfied if a is large enough. Therefore, by combining Theorems.5,.8, and 3. we get the following result. Theorem 3.. There exists a > such that the solution u quenches in a finite time for a < a and the solution exists globally for a > a. REFERENCES [] C.Y. Chan and W.Y. Chan, Existence of classical solutions for degenerate semilinear parabolic problems, Appl. Math. Comput. (999), [] C. Y. Chan and P. C. Kong, Quenching for degenerate semilinear parabolic equations, Appl. Anal. 54 (994), 7-5. [3] C. Y. Chan and H. T. Liu, Does quenching for degenerate parabolic equations occur at the boundaries?, Dynam. Contin. Discrete Impuls. Systems (Series A) 8 (), -8. [4] M.S. Floater, Blow-up at the boundary for degenerate semiliear parabolic equations, Arch. Rat. Mech. Anal. 4 (99),

9 EXISTENCE, UNIQUENESS AND QUENCHING 559 [5] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 964. [6] G.S. Ladde, V. Lakshmikantham and A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, MA, 985. [7] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural ceva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 968.