On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition

Transcription

1 J. Math. Anal. Appl. 286 (2003) On critical Fujita exponents for the porous medium equation with a nonlinear boundary condition Wenmei Huang, a Jingxue Yin, b andyifuwang a, a Department of Applied Mathematics, Beijing Institute of Technology, Beijing 0008, China b Department of Mathematics, Jilin University, Changchun, Jilin 3002, China Received 7 September 2000 Submitted by William F. Ames Abstract We establish the critical Fujita exponents for the solution of the porous medium equation u t = u m, x R N, t>0, subject to the nonlinear boundary condition um / x = u p, x = 0, t>0, in multi-dimension Elsevier Inc. All rights reserved.. Introduction In this paper we determine the critical Fujita exponent concerned with the following initial-boundary value problem: u t = u m, x R N,t>0, (.) u(x, 0) = u 0 (x), x R N, (.2) um x = u p, x = 0, t>0, (.3) Research partially supported by the NNSF and the MOST of China (Grants #997004, 02507, ). * Corresponding author. address: yifu_wang@63.net (Y. Wang) X/$ see front matter 2003 Elsevier Inc. All rights reserved. doi:0.06/s x(03)

2 370 W. Huang et al. / J. Math. Anal. Appl. 286 (2003) where R N ={(x,x ) x R N,x > 0}, m>, p>0, and u 0 (x) is a nonnegative bounded function satisfying the compatibility condition um 0 (x) = u p x 0 (x), x = 0, (.4) and is locally supported near some point, namely, for some x 0 R N, suppu 0 B R (x 0 ) R N and u 0(x) 0. However, the last assumption is not a real requirement for deriving our results. What we want to show is that even for the initial datum u 0 (x) vanishing except for a small ball, the solutions may still blow up in a finite time. The concept of critical Fujita exponents was proposed by Fujita before 970 s in discussing the heat conduction equation with nonlinear source; see, for example,. Following the idea of Fujita, we may define similar concepts for problem (.) (.3). We call p 0 the critical global existence exponent if it has the following property: if p>p 0, there always exist nonglobal solutions of problem (.) (.3) while if 0 <p<p 0,every solution of problem (.) (.3) is global. p c is called the critical Fujita exponent if for p 0 <p<p c any nontrivial solutions of problem (.) (.3) blows up in a finite time; for p>p c small data solutions exist globally in time while large data solutions are nonglobal. The problem of determining critical Fujita exponent is an interesting one in the general theory of blowing-up solutions to different nonlinear evolution equations of mathematical physics. Over the past few years there have been a number of extensions of Fujita result in various directions; see 2 6. Recently, it was Galaktionov and Levine 2 who first studied the one-dimensional case for the nonlinear boundary-value problem (.) (.3) with u 0 (x) having compact support. They showed that p 0 = (m )/2, p c = m. As for the similar questions with positive initial data u 0 (x), we refer to 7 9. This paper can be thought of as a natural continuation of 2 to multi-dimensional case. Because we are interested in the phenomena that local initial perturbation may cause blow up of solutions, we need not to consider the domain rather than the half-space R N.In fact, for a general domain, we may localize and flatten the boundary, and then make small modification for the arguments presented in this paper. The main result of this paper is that p 0 = (m )/2, p c = m /N. The idea of the proof is to construct super-solutions and sub-solutions inspired by 2. However, the subsolutions are quite different from those adopted in 2, since they should be chosen to have compact support in any spatial direction. For the process of verifying such kind of sub-solutions, we use the discriminant of cubic algebraic equations. 2. The main results and their proofs We need the following simple lemma. Lemma 2. (The discriminant of cubic algebraic equations). For the cubic equation x 3 px q = 0, (2.) there exist three roots x = A B, x 2,3 = A B ± i A B 3, 2 2

3 W. Huang et al. / J. Math. Anal. Appl. 286 (2003) where A = 3 q 2 Q, B = 3 q 2 Q, and Q = (p/3) 3 (q/2) 2. If Q = 0, (2.) has three real roots and x 2 = x 3 ; If Q>0, (2.) has one real root and two conjugate complex roots; If Q<0, (2.) has three unequal real roots. Theorem 2.2. If p>m /N, then any nontrivial nonnegative solution of the problem (.) (.3) blows up in finite time for large u 0. Proof. We begin with the construction of a nonglobal sub-solution of the self-similar form u(x, t) = (T t) 2p(m) θ(η), (2.2) where T>0isagiven constant, x η = ζ, ζ = pm. (T t) 2p(m) We can see that u(x, t) is a sub-solution of (.) (.3) if the function θ(η) satisfies ( η N η N (θ m ) ) p m 2p (m ) ηθ θ 0 (2.3) 2p (m ) for η {η>0 θ(η)>0} and θm θ p, ζ = 0. (2.4) ζ We claim that (2.3) and (2.4) admits a solution of the form θ(η)= A(a η) m (η b) m, 0 <b<η<a, for some positive constants A,a,b specified later. First, such θ(η) satisfies (2.4), since x = 0 implies ζ = 0and θm = θm η = θm ζ ζ η ζ η η implies θm ζ = 0 θ p. ζ =0 To check (2.), by a direct calculation we get θ =A m (a η) m (η b) m A (a η) m m (η b) m,

4 372 W. Huang et al. / J. Math. Anal. Appl. 286 (2003) (θ m ) = A m m m m (a η) m (η b) m m m m m (a η) (η b) m m m m (a η) (η b) m m (a η) m m (η b) m Substituting those into (2.3) and multiplying (2.3) by (a η) /(m) (η b) /(m), we have Set A m m (m ) 2 (η b)2 2Am m 2 (m ) 2 (a η) (η b) Am m (m ) 2 (a η)2 N A m m η (m ) (a η) (η b) 2 N A m m η (m ) (a η)2 (η b) p m 2p (m ) η A m (η b) A m (a η) e = A 2p (m ) (a η) (η b) 0. 2Am m (m ) 2 2Am m 2 (m ) 2 (N m )2Am m A m, e 2 = 2Am m (m ) 2 (a b) 2Am m 2 (m ) 2 (a b) (N ) Am m 3(a b) m p m A A (a b) (a b), 2p (m ) m 2p (m ) e 3 = Am m (m ) 2 (a2 b 2 ) 2Am m 2 (m ) 2 ab A 2p (m ) ab (N ) Am m m (a2 b 2 4ab), e 4 =(N ) Am m ab(a b). m We observe that (2.3) holds if e η 3 e 2 η 2 e 3 η e 4 0, 0 <b<η<a. (2.5) If we choose A large enough, it reduces to show where ẽ η 3 ẽ 2 η 2 ẽ 3 η ẽ 4 0, 0 <b<η<a, (2.6).

5 ẽ = 2 2m m W. Huang et al. / J. Math. Anal. Appl. 286 (2003) (N ), ẽ 2 = 2 2m (a b) 3(N )(a b), m ẽ 3 = m (a2 b 2 ) 2m m ab (N )(a2 b 2 4ab), ẽ 4 =(N )ab(a b). To do this, set y(η) =ẽ η 3 ẽ 2 η 2 ẽ 3 η ẽ 4. Notice that ẽ > 0 implies lim η y(η) =. Letting a = cb, c>, we want to show that if c, (2.6) holds. From Lemma 2. we know that when c, the equation y(η) = 0 has one real root and two conjugate complex roots. We only need to show that if c, y(b) 0, namely, 2 2m y(b) = 2(N ) b 3 (2 2m)( c) 3(N )( c) b 3 m m m ( c2 ) (N )(c 2 4c) 2m m c b 3 Let (N )c(c ) b m g(c) = m (2 2m)( c) 2(N ) m m ( c2 ) (N )(c 2 4c) 3(N )( c) 2m m c (N )c(c ) ; then y(b) = g(c)b 3,whereg(c) = ( c) 2 /(m ) 0. So y(b) 0. Thus we have verified that u(x, t) is a weak sub-solution of (.) (.3) and that u(x, t) blows up in a finite time. If for any given T, A, a, b, andc satisfying (2.3) and (2.4), the initial function u 0 is large enough such that u 0 (x) u(x, 0), x R N, then from the comparison principle (see 0), u(x, t) u(x, t) in R N (0,T) and hence u(x, t) blows up in a finite time which is not larger than T. The proof is complete. Theorem 2.3. If (m )/2 <p<m /N, then any nontrivial nonnegative solution of problem (.) (.3) blows up in finite time. Proof. We now use the idea used for a different problem in. We first notice that (.) (.3) admits the following well-known self-similar solution u B (x, t) = (τ t) N N(m)2 θ(η), (2.7)

6 374 W. Huang et al. / J. Math. Anal. Appl. 286 (2003) where x η = ζ, ζ =, (τ t) N(m)2 with τ>0anarbitrary constant. We can see that u(x, t) is a sub-solution of (.) (.3) if θ(η) satisfies ( η N η N (θ m ) ) N N(m ) 2 ηθ N(m ) 2 θ = 0 (2.8) and θ m ζ = 0. (2.9) ζ =0 Here η {η>0 θ(η) 0}. By a simple calculation, we see that θ(η)= A(c 2 η 2 ) /(m),0<η<c, satisfies (2.8) and (2.9), where { } m m A =. 2mN(m ) 2 Thus u B is a sub-solution to problem (.) (.3). By using the properties of weak solutions of problem (.) (.3), we deduce that there exist t 0 0 such that u(0,t 0 )>0. Since u(x, t 0 ) is a continuous function, there exist τ>0 large enough and small c>0such that u(x, t 0 ) u B (x, t 0 ), x R N. Then by comparison principle we deduce that u(x, t) u B (x, t), t t 0,x R N. (2.0) We now prove that there exist t t 0 and T large enough so that u B (x, t ) u(x, 0), x R N, (2.) where u(x, t) is the sub-solution given by (2.2). By using the space time structure of both functions u B and u, we choose suitable constants a,b such that 0 <ab<. If (τ t ) N N(m)2 T 2p(m) (2.2) and N (τ t ) N(m)2 T pm 2p(m) (2.3) are satisfied, (2.) is valid. We can see from (2.2) and (2.3) that such t and T exist if T 2p(m) T N(pm) 2p(m)

7 W. Huang et al. / J. Math. Anal. Appl. 286 (2003) for arbitrarily large T. This implies that N(p m) > 2p (m ) 2p (m ), namely, p<m N. Hence, from (2.0) and (2.), using the comparison principle we have that if p 0 <p <p c, u(x, t) blows up in a finite time. The proof is complete. Theorem 2.4. If p>m /N, then any nontrivial nonnegative solution of problem (.) (.3) is global in time for small u 0. Proof. We shall seek a global super-solution of the self-similar form ū(x, t) = (T t) 2p(m) Bg(η), (2.4) where η = ζ, x b x i ζ = pm, ζ i = pm (T t) 2p(m) (T t) 2p(m) (i = 2,...,N), T>0isagiven positive constant. We can see that ū(x, t) is a super-solution of (.) (.3) if g(η) 0 satisfies η N ( η N (g m ) ) p m 2p (m ) ηg g 0 (2.5) 2p (m ) and B m gm ζ B p g p. (2.6) ζ =b Here ζ {η>0 g(η) 0}, B>0. Nowweshowthatg(η) = A(c 2 η 2 ) /(m),where { A = m 2mN(m ) 2 } m, satisfies (2.5) and (2.6), where b (0,c).Using ( η N η N (g m ) ) N = N(m ) 2 ηg N(m ) 2 g, we see that g satisfies } B { m N N(m ) 2 ηg N(m ) 2 g p m 2p (m ) ηg g 0, 2p (m )

8 376 W. Huang et al. / J. Math. Anal. Appl. 286 (2003) namely, { 2 m B m N(m ) 2 p m 2p (m ) NB m } η 2 2p (m ) N(m ) 2 2p (m ) NB m N(m ) 2 c 2 0. Since p>m /N, we can choose a suitable constant B such that B m N(m ) 2 < p m 2p (m ), 2p (m ) < NB m N(m ) 2. Thus (2.5) is valid. Finally, we notice that, for g, inequality (2.6) is equivalent to (BA) pm (c 2 η 2 ) p m 2mb, m b<η<c. (2.7) If the inequality (BA) pm (c 2 b 2 ) p m 2mb m, b<η<c, holds, (2.7) is true. Setting c = ab, α, by choosing B small enough, we have (BA) pm (α 2 ) p m b 2p(m) m 2m m, which implies that (2.7) is valid. Thus, for p>p c, there exists a nontrivial global super-solution, and hence a class of small global solutions. We have thus completed the proof. Theorem 2.5. If 0 <p<p 0, then any nontrivial nonnegative solution of problem (.) (.3) is global in time. Proof. If p = p 0 = (m )/2, we can construct a global super-solution of the self-similar form u (x,x,t)= u (x, 0,t)= e α(t t) h(ζ ), where ζ = x /e α(tt)/2, T>0isagiven positive constant, and α>0. We can see that u (x, t) is a solution of (.) (.3) if h(ζ ) satisfies (h m ) α(m ) (ζ ) ζh (ζ ) αh(ζ) = 0 (2.8) 2 and (h m ) (0) = h p 0 (0). (2.9) From 2 we see that there exists a unique solution h 0 which has compact support on x such that (2.8) and (2.9) hold. Thus, we can choose T large enough such that u 0 (x) u (x, 0), x R N.

9 W. Huang et al. / J. Math. Anal. Appl. 286 (2003) Using comparison principle, we have u(x, t) u (x, t). From the global existence of u (x, t), we see that u(x, t) is also global in time. If p<p 0, u (x, t) is a global super-solution of (.) (.3) whenever u (0,t). Hence, using comparison principle again, we can get the global existence of u(x, t). The proof is complete. References H. Fujita, On the blowing up of solutions of the Cauchy problem for u t = u u α, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 3 (966) V.A. Galaktionov, H.A. Levine, On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math. 94 (996) H.A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev. 32 (990) M. Escobedo, M.A. Herrero, Boundedness and blow up for a semi-linear reaction diffusion system, J. Differential Equations 89 (99) B. Hu, H.M. Yin, On critical exponents for the heat equation with a nonlinear boundary condition, Ann. Inst. H. Poincaré 3 (996) K. Deng, H.A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl. 243 (2000) M.X. Wang, Y.H. Wu, Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, Trans. Amer. Math. Soc. 349 (997) M.X. Wang, S.X. Chen, Convective porous medium equations with nonlinear forcing at the boundary, J. Math. Anal. Appl. 2 (997) J.S. Hu, M.X. Wang, Existence and nonexistence of global solutions of the convective porous medium equations with a nonlinear forcing at the boundary, J. Math. Anal. Appl. 97 (996) M.X. Wang, Nonlinear Parabolic Equations, Scientific Press, Beijing, 993. V.A. Galaktionov, S.P. Kurdyunov, A.P. Mikhailov, A.A. Samarskii, On blowing-up solutions to the Cauchy problem for the parabolic equation u t = (u α u) u β, Dokl. Akad. Nauk SSSR Ser. Math. Phys. 252 (980) , English transl. Soviet Phys. Dokl. 25 (980) B.H. Gilding, L.A. Peletier, On a class of similarity solutions of the porous media equation, J. Math. Anal. Appl. 55 (976)