Critical exponent for semilinear wave equation with critical potential

Transkript

1 Nonlinear Differ. Equ. Appl. (13), c 1 Springer Basel 11-97/13/ published online December 16, 1 DOI 1.17/s x Nonlinear Differential Equations and Applications NoDEA Critical exponent for semilinear wave equation with critical potential Xinfu i Abstract. We consider the Cauchy problem for the semilinear wave equation u tt Δu + V (x)u t = u p. When V (x) =V (1 + x ) 1/, V n, we prove that the critical exponent for the problem is p c(n) =, n, n , n =1. Mathematics Subject Classification (1). 355; 357. Keywords. Damped wave equation, Critical exponent, Blow up, Global existence. 1. Introduction We consider the Cauchy problem for the semilinear wave equation utt Δu + V (x)u t = u p, (x, t) (, ), u(x, ) = ɛu (x), u t (x, ) = ɛu 1 (x), x (1.1), where ɛ>, psatisfies 1 <p<+ (n =1, ), 1 <p n (n 3), n (u,u 1 ) H 1 satisfy: there exists a constant R >, such that suppu,u 1 } B R () = x x <R}, and V (x) C( ) is a potential function which will be specified later. We focus on the critical exponent p c (n) of the problem (1.1), which is a number defined by the following property: If p>p c (n), then all solutions of (1.1) with small initial values are global; while if 1 <p p c (n), then all solutions of (1.1) with nonnegative initial values blow up in finite time regardless of the smallness of the initial values. When the potential V (x) is a constant (V (x) 1), Todorova and Yordanov [8] in 1 obtained that the critical exponent of (1.1) is1+ n, which is

2 138 X. i NoDEA the same as the Fujita exponent for the heat equation v t Δv = v p (see Fujita [1] and Weissler [1]). More precisely, they proved that: If 1 + n <p n n (n 3), 1+ n <p< (n =1, ), and the initial values with compact supports are sufficiently small, then the problem (1.1) admits a unique global solution u C([, ),H 1 ) C 1 ([, ), ); While if 1 <p<1+ n, u (x) dx >, u 1 (x) dx >, then all the solutions to (1.1) blow up in finite time. ater Zhang [11roved that the critical exponent 1 + n belongs to the blow up region. In 5, Ikehata and Tanizawa [] extended the global existence result of [8] to initial values without compact supports. When V (x) C 1 ( ) is a radially symmetric function, V (x) V (1 + x ) α, x, V >, α (, 1), Ikehata et al. [3] in 9 obtained that the critical exponent for (1.1)is1+ n α. More precisely, they proved that: If 1 + n+ n α <p< n (n 3), 1+ n α <p< (n =1, ), and the initial values are sufficiently small, then the problem (1.1) admits a unique global solution u C([, ),H 1 ) C 1 ([, ), ), ( u t + u ) dx C(1 + t) ( n α α +1 δ), where δ> is an arbitrarily small number; While if 1 <p 1+ n α, (u 1 + V (x)u ) dx >, then all the solutions to (1.1) do not exist globally for any ɛ>. In this paper we solve the critical exponent problem for (1.1) with critical potential V (x). More precisely, we assume that V (x) C( ) satisfies: V (1 + x ) 1/ V (x) V 1 (1 + x ) 1/, V,V 1 >. (1.) To get the critical exponent we need the decay estimates for the homogeneous problem utt Δu + V (x)u t =, (x, t) (, ), u(x, ) = u (x), u t (x, ) = u 1 (x), x (1.3), where V (x) satisfies (1.), and (u,u 1 ) H 1, suppu,u 1 } B R (). Matsumura [5], Mochizuki and Nakazawa [6], Uesaka [9] discussed the energy decay rate of the problem (1.3), and obtained (u t + u ) dx C(1 + t) minv,1}. Recently, Ikehata et al. [4] obtained that the solution to (1.3) satisfies: when n =1,, (u t + u Cδ I ) dx (1 + t) V+δ, <V n, R CI n (1 + t) n, V >n,

3 Vol. (13) Critical exponent for semilinear wave equation 1381 when n 3, C δ I (1 + t) V+δ, <V 1, (u t + u ) dx CI (1 + t) V, 1 <V <n, R n C δ I (1 + t) n+δ, V n, where I = u H 1 + u 1, δ > is an arbitrarily small number. Using multiplier method, in another paper, we can prove that when n =, (u t + u ) dx CI (1 + t) V, 1 <V. Above all, the optimal estimates obtained for the solutions to (1.3) are as follows: when n =1,, (u t + u ) dx CI (1 + t) minv,n}, (1.4) when n 3, (u t + u ) dx CI (1 + t) V, <V <n, C δ I (1 + t) n+δ, V n, (1.5) where δ> is an arbitrarily small number. Now we are ready to state our main results. The main result of this paper is that the critical exponent p c (n) for(1.1) (V n) isp c (n) =1+ n 1. Our global existence result is as follows. Theorem 1.1. et n, V(x) satisfy (1.), V n, 1+ n 1 <p<+ (n = ), and 1+ n 1 <p n n (n 3). Then there exists ɛ > such that problem (1.1) admits a unique global solution u C([, ),H 1 ) C 1 ([, ), ), for each ɛ<ɛ. Moreover, for any δ (, min 1 4, (n 1)p (n+1) (p 1) }], the global solution satisfies Du Cɛ( u H 1 + u 1 )(1 + t) 1, n =, Du Cɛ( u H 1 + u 1 )(1 + t) n +δ, n 3, where D =( t, x ), C > is a constant, independent of t. The blow up result is as follows. Theorem 1.. et V (x) satisfy (1.), 1 < p < (n = 1), and 1 < p 1+ n 1 (n ). If V (x)u 1(x)+u (x) and V (x)u 1 (x)+u (x), then the solution to (1.1) does not exist globally, for any ɛ>. Remark 1.3. We did not find good methods to solve the critical exponent problem to (1.1) for<v <n. This paper is organized as follows. In Sect. we prove Theorem 1.1 by dividing the proof into several lemmas. In Sect. 3 we prove Theorem 1..

4 138 X. i NoDEA. Global existence result for small initial values The following local result for the problem (1.1) is well-known, which can be obtained by a simply modification of the result in Strauss [7]. emma.1. et V (x) C( ) satisfy (1.). Then the problem (1.1) admits a unique local solution u C([,T),H 1 ) C 1 ([,T), ) satisfying u(x, t), x t + R, where T>depending only on Du(). Moreover, the solution can be continued beyond the interval [,T) if sup Du(t) < +. [,T ) In view of this local existence result, global existence of a solution follows the boundedness of the energy at all times. In the following, we prove Theorem 1.1 by the method of Todorova and Yordanov [8]. Choosing a weight function ψ(x, t) = 1+ x 1+t, we obtain the following weighted energy estimate. emma.. et V (x) satisfy (1.), V 1, u(x, t) be a local solution to (1.1) on [,T), and ψ(x, t) = 1+ x 1+t. Then for any t [,T), the following estimate holds: ( ) ()/ e ψ Du Cɛ + C max (s e γ1ψ(s) u(s) [,t] +1)η1, where η 1 >, γ 1 > independent of ɛ and t. are arbitrarily numbers, and C> is a constant, Proof. Multiplying Eq. (1.1) bye ψ u t and rearranging the terms, we obtain (e ψ u t + u ) (e ψ u t u)+ eψ ψ t u u t ψ t ψ t ( + eψ e (ψt ψ V (x)ψ t )u ψ u p ) u t = eψ u p uψ t. (.1) ψ t p +1 p +1 By the choice of ψ(x, t) andv 1, we obtain ψ t ψ V (x)ψ t ψ t = 1+ x (1 + t) >, ψ x i = 1 1+t x i t x, ψ = 1 (1 + t), (1 + x ) (1 + t) 4 1 (1 + t) + 1+ x (1 + t) V 1+ x. Thus, by the above arguments, (.1) can be simplified to (e ψ u t + u ) ( e (e ψ ψ u p ) u u t u) eψ u p uψ t. (.) t p +1 t p +1

5 Vol. (13) Critical exponent for semilinear wave equation 1383 Integrating (.) over[,t], and letting ɛ 1, we obtain t e ψ Du C(ɛ + ɛ )+C e ψ u dx + C e ψ u ψ s dxds Cɛ + C e ψ u t ( +C ψ s e ( γ1())ψ(s)) e γ1ψ(s) u(s) ds, max suppu(s) where γ 1 >, and C > is a constant, independent of ɛ and t. By the choice of ψ(x, t), we have max ψ s e ( γ1())ψ(s) = suppu(s) max x s+r 1+ x (1 + s) e( γ1())ψ(s) 1+R + s (1 + s) C 1+s. Thus, we have t e ψ Du Cɛ + C e ψ u + C 1 1+s eγ1ψ(s) u(s) ds Cɛ + C e ψ u [max + C (1 + e γ1ψ(s) u(s) [,t] s)η1 [ ] Cɛ + C max (1 + e γ1ψ(s) u(s) [,t] s)η1, where η 1 > is an arbitrarily number. We complete the proof of emma.. ] emma.3. et n, V(x) satisfy (1.), V n, u(x, t) be a local solution to (1.1) on [,T), and δ (, 1 4 ] is any fixed number. Then for any t [,T), the following estimates hold: when n =, when n 3, Du(t) C(1 + t) 1 (ɛ + max [,t] Du(t) C δ (1 + t) n/+δ (ɛ + max [,t] where η> and γ> are arbitrarily small numbers. [(1 + s) 1+η p e γψ(s) u(s) p ), [(1 + s) n/ δ p e γψ(s) u(s) p ), Proof. et u (x, t) be the solution to the homogeneous problem utt Δu + V (x)u t =, (x, t) (, ), u(x, ) = ɛu (x), u t (x, ) = ɛu 1 (x), x, and S(t) u 1 (x) be the solution to the problem utt Δu + V (x)u t =, (x, t) (, ), u(x, ) =, u t (x, ) = u 1 (x), x. (.3)

6 1384 X. i NoDEA Then u(x, t) =u (x, t)+ t S(t τ) u p (τ) dτ is a solution to the problem (1.1). In the following, we split the proof of emma.3 into two cases. Case 1 (n = ). By the energy estimate (1.4) for problem (.3), the linear term Du (t) is bounded by Du (t) Cɛ(1 + t) 1 ( u H 1 + u 1 ) Cɛ(1 + t) 1, (.4) and the integral term is estimated as follows: ( t t D S(t τ) u p (τ) dτ) DS(t τ) u p (τ) dτ C t (1 + t τ) 1 u p (τ) dτ = C Since ψ(x, t) >, for any γ>, we have t (1 + t τ) 1 u(τ) p p dτ. (.5) u(τ) p e γψ(τ) u(τ) p. (.6) Inserting (.6) into(.5) and splitting the integral into two parts, we have ( t D S(t τ) u p (τ) dτ) ( t/ ) t C + (1 + t τ) 1 e γψ(τ) u(τ) p p dτ = C(I 1 + I ). (.7) For any η>, t/ t/ I 1 = (1 + t τ) 1 1 (1 + τ) 1+η (1 + τ)1+η e γψ(τ) u(τ) p p dτ [ C(1 + t) 1 max (1 + τ) (1+η)/p e γψ(τ) u(τ) p, (.8) t [,t/] I = (1 + τ) 1 (1 + t τ) (1+η) (1 + t τ) η (1 + τ) e γψ(τ) u(τ) p p dτ t/ [ C(1 + t) 1 max (1 + τ) (1+η)/p e γψ(τ) u(τ) p. (.9) [t/,t] Combining (.4), (.7), (.8) and (.9), we obtain, when n =, ] ) Du(t) C(1 + t) (ɛ 1 + max [(1 + s) 1+η p p e γψ(s) u(s) p. [,t] Case (n 3). By the energy estimate (1.5) for problem (.3), the linear term Du (t) is bounded by Du (t) C δ ɛ( u H 1 + u 1 )(1 + t) n/+δ, (.1)

7 Vol. (13) Critical exponent for semilinear wave equation 1385 and the integral term is estimated as follows: ( t t D S(t τ) u p (τ) dτ) DS(t τ) u p (τ) dτ t t C δ (1+t τ) n/+δ u p (τ) dτ =C δ (1+t τ) n/+δ u(τ) p p dτ ( t/ ) t C δ + (1+t τ) n/+δ e γψ(τ) u(τ) p p dτ =C δ(i 3 +I 4 ). (.11) For any η>, t/ t/ I 3 = (1 + t τ) n/+δ 1 (1 + τ) 1+η (1 + τ)1+η e γψ(τ) u(τ) p p dτ [ C(1 + t) n/+δ max (1 + τ) (1+η)/p e γψ(τ) u(τ) p, (.1) [,t/] t I 4 = (1 + τ) n/+δ (1 + t τ) n/+δ (1 + τ) n/ δ e γψ(τ) u(τ) p p dτ t/ [ C(1 + t) n/+δ max (1 + τ) (n/ δ)/p e γψ(τ) u(τ) p. (.13) [t/,t] Combining (.1) (.13), we obtain, when n 3, Du(t) C δ (1 + t) n/+δ (ɛ + max [,t] [(1 + s) n/ δ p e γψ(s) u(s) p ), which completes the proof of emma.3. emma.4. et σ (, 1], q< (n =), q n (n 3), ψ(x, t) = 1+ x 1+t,u(x, t) be a local solution of (1.1) on[,t). Then for any t [,T), the following estimate holds: e σψ(t) u q C(1 + t) 1 θ(q) e ψ(t) u σ u 1 σ, where θ(q) =n( 1 1 q ), and C> is a constant, independent of t. Proof. Applying the Gagliardo Nirenberg inequality to e σψ u, we have e σψ u q e σψ u 1 θ(q) (e σψ u) θ(q). (.14) On the other hand, e σψ u = (e σψ u) σe σψ u ψ. Thus, e σψ u = ( (e σψ u) + σ e σψ u ψ σe σψ u (e σψ u) ψ) dx R n = ( (e σψ u) + σ e σψ u ψ + σδψ(e σψ u) ) dx. (.15) By the choice of ψ(x, t), we have ψ 1 = (1 + t), Δψ = n t x. (.16) n

8 1386 X. i NoDEA Inserting (.16) into(.15), we have e σψ u (e σψ u) + By (.17), (.14) can be simplified to σ (1 + t) eσψ u. (.17) e σψ u q C(σ)(1 + t) 1 θ(q) e σψ u. (.18) Using Hölder inequality, e σψ u = e σψ u σ u (1 σ) dx R ( n ) σ ( ) 1 σ e ψ u u. Thus, e σψ u e ψ u σ u 1 σ. (.19) Inserting (.19) into(.18), we complete the proof of emma.4. Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. We split the proof into two cases. Case 1 (n = ). We introduce the weighted energy functional By emmas. and.3, wehave W (t) Cɛ + C W (t) = e ψ Du +(1+t) Du. [ +C ( max [,t] (s +1)η1 e γ1ψ(s) u(s) 1+η max(1 + s) [,t] ) ()/ p e γψ(s) u(s) p, (.) where η>, γ >, η 1 >, γ 1 > are arbitrarily small numbers. By emma.4 and the definition of W (t), we obtain e γ1ψ(s) u(s) C(1 + s) 1 θ() e ψ(s) u(s) γ1 u(s) 1 γ1 C(1 + s) 1 θ() (1 γ1) W (s), (.1) e γψ(s) u(s) p C(1 + s) 1 θ(p) e ψ(s) u(s) γ u(s) 1 γ C(1 + s) 1 θ(p) (1 γ) W (s), (.) where θ(p +1)=n( 1 1 ),θ(p) =n( 1 1 p ). Using (.1) and (.), we obtain from (.) W (t) Cɛ + C max [,t] (s +1) η1+ [1 θ() (1 γ1)] W (s) ()/ +C max [,t] (1 + s)1+η+p[1 θ(p) (1 γ)] W (s) p. (.3)

9 Vol. (13) Critical exponent for semilinear wave equation 1387 In the following we calculate the exponents of (1 + s) in(.3). Set γ 1 = + η,η >, then p +1 η 1 + p +1 [1 θ(p +1) (1 γ 1)] = p +1 (η 1 + η )+ 3 p, 1+η + p[1 θ(p) (1 γ)] = η + pγ +( p). Since p>3, we can choose η>, λ >, η 1 >, η > sufficiently small, such that the exponents of (1 + s) are negative. Thus we have from (.3) W (t) Cɛ + C max W (s) + C max W [,t] [,t] (s)p. (.4) Set M(t) = max [,t] W (s). From (.4), we have M(t) Cɛ, for sufficiently small ɛ, which completes the proof of Theorem 1.1 for n =. Case (n 3). For any fixed δ (, min 1 4, (n 1)p (n+1) (p 1) }], we introduce the weighted energy functional By emmas. and.3, wehave W 1 (t) = e ψ Du +(1+t) n δ Du. ( ) ()/ W 1 (t) Cɛ + C max (s e γ1ψ(s) u(s) [,t] +1)η1 n/ δ +C δ [max(1 + s) p e γψ(s) u(s) p, (.5) [,t] where γ>, η 1 >, γ 1 > are arbitrarily small numbers. By emma.4 and the definition of W 1 (t), we obtain e γ1ψ(s) u(s) C(1 + s) 1 θ() e ψ(s) u(s) γ1 u(s) 1 γ1 C(1 + s) 1 θ() (n/ δ)(1 γ1) W 1 (s), (.6) e γψ(s) u(s) p C(1 + s) 1 θ(p) e ψ(s) u(s) γ u(s) 1 γ C(1 + s) 1 θ(p) (n/ δ)(1 γ) W 1 (s), (.7) where θ(p +1)=n( 1 1 ),θ(p) =n( 1 1 p ). Using (.6) and (.7), we obtain from (.5) W 1 (t) Cɛ + C max [,t] (s +1) η1+ [1 θ() (n/ δ)(1 γ1)] W 1 (s) ()/ +C max [,t] (1 + s)n/ δ+p[1 θ(p) (n/ δ)(1 γ)] W 1 (s) p. (.8)

10 1388 X. i NoDEA In the following we calculate the exponents of (1 + s) in(.8). Set γ 1 = + η,η >, then p +1 η 1 + p +1 [1 θ(p +1) (n/ δ)(1 γ 1)] = p +1 (η 1 + η (n/ δ)) + p 1 δ + 1 [n +1 (n 1)p], n/ δ + p[1 θ(p) (n/ δ)(1 γ)] =(n/ δ)γp +(p 1)δ +[n (n 1)p]. Since p>1+ n 1,δ (, min 1 4, (n 1)p (n+1) (p 1) }], we have p 1 δ + 1 [n +1 (n 1)p] <, (p 1)δ +[n (n 1)p] <. Choose γ>, η 1 >, η > sufficiently small, such that the exponents of (1 + s) are negative. Thus we have from (.8) W 1 (t) Cɛ + C max W 1(s) + C max W 1(s) p. (.9) [,t] [,t] Set M 1 (t) = max [,t] W 1 (s). From (.9), we have M 1 (t) Cɛ, for sufficiently small ɛ, which completes the proof of Theorem 1.1 for n Blow up result In this section we prove Theorem 1. by the method of test functions (see Zhang [11]). Proof of Theorem 1.. Choose a function φ(s) C ([, )) satisfying 1, s < 1, φ(s) = φ(s) 1,, s >, φ (s) Cφ 1 p (s), φ (s) Cφ 1 p (s), where C>isaconstant. Set ϕ(x, t) =φ( x )φ(t). For any >, define ( x ϕ (x, t) =ϕ, t ), (x, t) [, ). In the following, we prove Theorem 1. by contradiction. Suppose u(x, t) C([, ),H 1 ) C 1 ([, ), ) is a global solution to (1.1). Multiplying equation (1.1) byϕ (x, t) and integrating the corresponding equality by parts over [, ), we obtain u( tt ϕ Δϕ V (x) t ϕ ) dxdt R n (u 1 (x)ϕ (x, ) u (x) t ϕ (x, ) + V (x)u (x)ϕ (x, )) dx R n = ϕ u p dxdt. (3.1)

11 Vol. (13) Critical exponent for semilinear wave equation 1389 By the choice of ϕ (x, t), we have ( ) x ϕ (x, ) = φ, t ϕ (x, ). By the assumption u 1 (x)+v(x)u (x), we obtain from (3.1) ϕ u p dxdt u( tt ϕ Δϕ V (x) t ϕ ) dxdt. (3.) Using Hölder inequality, we estimate the integral on the right in (3.) u( tt ϕ Δϕ V (x) t ϕ ) dxdt R ( n ) 1/p C u p ϕ dxdt + u p ϕ dxdt ( B () ϕ p /p B ()\B () tt ϕ Δϕ V (x) t ϕ p dxdt ) 1/p where p = p/(p 1). By the properties of ϕ (x, t), we obtain ϕ p /p tt ϕ Δϕ V (x) t ϕ p dxdt R n ( = ϕ p /p x R, t ) ( 1 x ϕ tt, t ) 1 ( x Δϕ, t ) n V (x) 1 ( x ϕ t, t ) p dxdt C dxdt + C V (x) p dxdt p p B () B () (1 + r) n 1 C n+1 p + C 1 p dr (1 + r) p C n+1 p, n p >, C 1 p ln, n p =, C 1 p, n p <. Inserting (3.3) and (3.4) into(3.), we obtain ( ) p/(p 1) ϕ u p dxdt R ( n C u p ϕ dxdt + B () C n+1 p, n p >, C 1 p ln, n p =, C 1 p, n p <. B ()\B () u p ϕ dxdt Finally, we show that the above inequality can not hold as., (3.3) (3.4) ) 1/(p 1) (3.5)

12 139 X. i NoDEA Case 1. When 1 <p<1+ n 1 (n ), 1 <p< (n =1), the exponents of in (3.5) are negative. In (3.5), letting, we obtain u p dxdt, which is impossible, since u is a nontrivial solution. Case. When p =1+ n 1 (n ), the exponents of in (3.5) are nonpositive. In (3.5), letting T, we obtain u p dxdt C. (3.6) In (3.5), letting T, and considering (3.6), we obtain u p dxdt, which is impossible, since u is a nontrivial solution. By Case 1 and Case, we complete the proof of Theorem 1.. References [1] Fujita, H.: On the blowing up of solutions to the Cauchy problem for u t = Δu + u 1+α,.J.Fac.Sci.Univ.TokyoSect.I13, (1966) [] Ikehata, R., Tanizawa, K.: Global existence of solutions for semilinear damped wave equations in R N with noncompactly supported initial data. Nonlinear Anal. 61, (5) [3] Ikehata, R., Todorova, G., Yordanov, B.: Critical exponent for semilinear wave equations with space-dependent potential. Funkcialaj Ekvacioj 5, (9) [4] Ikehata, R., Todorova, G., Yordanov, B.: Optimal decay rate of the energy for wave equations with critical potential (1, preprint) [5] Matsumura, A.: Energy decay of solutions of dissipative wave equations. Proc. Japan Acad. Ser. A 53, 3 36 (1977) [6] Mochizuki, K., Nakazawa, H.: Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation. Publ. RIMS. Kyoto Univ. 3, (1996) [7] Strauss, W.A.: Nonlinear wave equations, in: CBMS Regional Conference Series in Math., vol. 73, AMS, Providence, ISBN: , 1989, x+91 pp. Published for the conference board of the Math. Sci. Washington, D.C. [8] Todorova, G., Yordanov, B.: Critical exponent for a nonlinear wave equation with damping. J. Differ. Equ. 174, (1) [9] Uesaka, B.: The total energy decay of solutions for the wave equation with a dissipative term. J. Math. Kyoto Univ. (1), (1979)

13 Vol. (13) Critical exponent for semilinear wave equation 1391 [1] Weissler, F.: Existence and non-existence of global solutions for a semilinear heat equation. Israel J. Math. 38, 9 4 (1981) [11] Zhang, Q.S.: A blow-up result for a nonlinear wave equation with damping: the critical case. C. R. Acad. Sci. Paris Sér. I 333, (1) Xinfu i School of Science Tianjin University of Commerce Tianjin 3134 China lxf13465@pku.edu.cn Received: 4 May 1. Accepted: 6 December 1.