NONLINEAR PSEUDODIFFERENTIAL EQUATIONS ON A HALF-LINE WITH LARGE INITIAL DATA

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1 Electronic Journal of Differential Equations, Vol. 26(26), No. 89, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) NONLINEAR PSEUDODIFFERENTIAL EQUATIONS ON A HALF-LINE WITH LARGE INITIAL DATA ROSA E. CARDIEL, ELENA I. KAIKINA Abstract. We study the initial-boundary value problem for nonlinear pseudodifferential equations, on a half-line, u t + λ u σ u + Lu =, (x, t) R + R +, u(x, ) = u (x), x R +, where λ > and pseudodifferential operator L is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions and to find the main term of the asymptotic representation in the case of the large initial data. 1. Introduction We consider the initial-boundary value problem for a general class of the nonlinear nonlocal equations, on a half-line, u t + λ u σ u + Lu =, (x, t) R + R +, u(x, ) = u (x), x R +, (1.1) where λ >, σ >. The linear operator L is a pseudodifferential operator defined by the inverse Laplace transform as follows where (1, 2) and Lu = 1 2πi i û(p) = e px C p ( û(p, t) e px u(x)dx u(, t) ) dp, (1.2) p denotes the Laplace transform of u. We assume that the symbol K(p) = C p is dissipative, i.e. R(K(p)) > for R(p) =. Here and below p is the main branch of the complex analytic function in the half-complex plane Rp, so that 1 = 1 (we make a cut along the negative real axis (, )). The initial-boundary value problem (1.1) is of great interest from the physical point of view, since it describes many physical phenomena, such as the focusing of laser beams, waves on water (some other applications can be found [28]). A great number of publications have dealt with asymptotic representations of solutions to 2 Mathematics Subject Classification. 35Q35, 35B4. Key words and phrases. Pseudodifferential operator; large data; asymptotic behavior. c 26 Texas State University - San Marcos. Submitted February 1, 26. Published August 9, 26. 1

2 2 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 the Cauchy problem for nonlinear evolution equations in the last twenty years. While not attempting to provide a complete review of this publications, we do list some known results [3]-[1],[12],[13],[15] -[18], [21], [22], [26]-[35], where there were obtained optimal time decay estimates and asymptotic formulas of solutions to different nonlinear local and nonlocal dissipative equations. The asymptotic theory of the initial-boundary value problems for the nonlinear pseudodifferential equations is relatively new and traditional questions of a general theory are far from their conclusion. A description of the large time asymptotic behavior of solutions for the initial-boundary value problems requires new approaches and the reorientation of the points of view compared to the Cauchy problem. The initial-boundary value homogeneous problems for nonlinear pseudodifferential equations were studied in the book [2]. In the present paper we continue the study of pseudodifferential equations on a half-line, considering the case of a large initial data. The aim of this paper is to prove a global existence of solutions to the initialboundary value problem (1.1), and to find the main term of the asymptotic representation of solutions. We will obtain the a priory optimal time decay estimates of solutions in the usual Lebesgue spaces L r for 1 r. These type of estimates enable us to consider the critical and sub critical cases in future works. To state our results we give some notations. The weighted Sobolev space is H k,s r = {f L r : f H k,s r where x = 1 + x 2. We introduce the function Λ (s) L, where Λ (s) = ( 1) {} C A 1 2π 2 sin( π{} i π A 1 ) = i x k x s f L r < }, i dze sz z {} dqe q K(z) + q, A 1 = ( 1) {}+1 ( C ) [] Γ(1 {})Γ({}) sin π{}. We prove the following theorem. q {} (1.3) Theorem 1.1. Let (1, 2), σ >,λ > and real valued function u H 1, 2 H,µ H, 1, where µ [, 1]. Then there exists a unique real valued solution of (1.1) such that u(x, t) C([, ); H 1, 2 H,µ H, 1 ). Moreover there exists a constant A such that u(x, t) = t 1/ AΛ (xt 1/ ) + O(t 1+µ ) (1.4) as t uniformly with respect to x >, where A = u dy + λ dt u σ u dy. Remark 1.2. We can guarantee that the coefficient A in the asymptotic representation (1.4) if u dy and u σ u is small. It can occur that A =, for instance, for convective equations u σ u dy if the initial data have zero mean value u dy =. In the last case formula (1.4) gives us only some time decay estimate for the solutions.

3 EJDE-26/89 NONLINEAR PSEUDODIFFERENTIAL EQUATIONS 3 2. Preliminaries We introduce the function κ(ξ) = K 1 ( ξ), such that Re κ(ξ) > for all Re ξ. We denote G(t 1 )φ(t 2 ) = where Green function G(x, y, t) is defined by G(x, y, t) = 1 2πi + 1 4π 2 i i e K(p)t+p(x y) dp G(x, y, t 1 )φ(y, t 2 ) dy, e ξt κ(ξ)ξ 1 e κ(ξ)y i e px K(p) dξ dp. p(k(p) + ξ) (2.1) The solution u of the problem (1.1) can be represented as follows (see [2], pp ) u(x, t) = G(t)u + t where N (u) = λ u σ u. We introduce complete metric space dτg(t τ)n (u)(τ), (2.2) X = {φ(x, t) C([, ); H,µ r ) C((, ); H 1, r ) φ X < +, }, φ X = sup t 1 ( t 1 r µ x µ φ L r + t 1 1 r φx L r), t> where µ [, 1], 1 r. Let a continuous linear functional f(φ) : L 1 R is defined as f(φ) = φ(x)dx. Now we obtain some estimates for the Green operator G in the space X. Lemma 2.1. The following two estimates are valid Gφ X C( φ L 1 + φ L ), (2.3) sup t 1 (1+µ) Gφ t 1 Λ (xt 1/ )f(φ) L C φ L 1,µ (2.4) t>1 for all t >, provided that the right-hand sides are bounded. Proof. We rewrite the Green function as where G(x, y, t) = F 1 (x y, t) + F 2 (x, y, t), i F 1 (x, t) = 1 e K(p)t+px dp (2.5) 2πi F 2 (x, y, t) = 1 i i 4π 2 e ξt k(ξ)y k(ξ)ξ 1 e px K(p) dξ dp. (2.6) p(k(p) + ξ) Firstly we prove the estimate t 1 1 r ( F 1 (q, t) L r + t µ q µ F 1 (q, t) L r + t 1 q F 1 (q, t) L r ) C (2.7) for all t >, where µ > and 1 r. Changing variables p t = z, and q = qt 1/ we get F 1 (q, t) = 1 i e qp e K(p)t dp i Ct 1/ e eqz e Cz dz. 2πi

4 4 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 Therefore, After the change p t = z, we have q µ F 1 (q, t) Ct 1 + µ i q µ q F 1 (q, t) Ct 2 where < γ <. Therefore, and F 1 (q, t) L r Ct r. i q µ F 1 (q, t) L r Ct 1 + µ + r 1 (( Ct 1 + µ + r 1 e eqz e Cz dz Ct 1 + µ q µ 1 γ, ze eqz e Cz dz Ct 2 q 1 γ, 1 Estimate (2.7) is then proved. Now we prove + 1 )d q(1 + q 2 ) (µ 1 γ)r 2 ) 1/r (2.8) q F 1 (q, t) L r Ct r. (2.9) sup t 1 1 r +µ1+γ t 2γ y µ1 ( F 2 (x, y, t) L r t>,y> + t µ x µ F 2 (x, y, t) L r + t 1 x F 2 (x, y, t) L r) C, (2.1) where µ, γ >, µ 1 < 1 (1 1 r γ) and 1 r. We have, by definition, κ(q) = C 1 q 1. Changing variables p t = z and ξt = q, we obtain the estimate F 2 (x, y, t) Ct 1/ i i dze exz K(z)z 1 dq We move the contours of integration with respect to z as follows and with respect to q by 1 q 1 κ(q) eq C1q ey. (2.11) K(z) + q C 1 = {z = ρe ±iβ1, ρ, β 1 = π 2 + ɛ 1} (2.12) C 2 = {q = e iφ2 φ 2 [ π 2, π 2 ]} {q = ρe±iφ2 ρ 1, φ 2 = π 2 + ɛ 2}, (2.13) where ɛ 1, ɛ 2 > are small enough. It is apparent that e q C q γ, K(z) + q 1 C z ν q ν 1, e z x C z x µ2, e C1jq 1 ey C q 1 ỹ µ 1 (2.14) for all q C 2 and z C 1, where ν [, 1], γ, and µ 1, µ 2. Taking into account (2.14) we get F 2 (x, y, t) Ct 1/ y µ1 x i i µ2 dz z 1 ν µ2 dq q 1 2+ν µ1 γ. (2.15) To guarantee the convergence of this integrals we need to satisfy the following conditions ν µ 1 > 1 (2.16)

5 EJDE-26/89 NONLINEAR PSEUDODIFFERENTIAL EQUATIONS 5 and 1 ν µ 2 > 1, [] ν µ 2 < 1. (2.17) respectively. Under the condition µ 1 + µ 2 < 2 [] there exists some ν such that the estimates (2.17) are valid. Therefore, we obtain From (2.18) it follows that F 2 (p, y, t) Ct 1 µ 2 +µ1 x µ2 y µ1. (2.18) F 2 (x, y, t) L r Ct r +µ1±γ y µ1( x µ2r dx ) 1/r Ct r +µ1±γ y µ1, where µ 1 < 1 (1 1 r ). In the same manner we obtain Thus we obtain x µ F 2 (x, y, t) Ct 1 + µ µ 2 +µ1 x µ µ2 y µ1, x F 2 (p, y, t) Ct 2 µ 2 +µ1 y µ1 x µ2. (2.19) x µ F 2 (x, y, t) L r Ct 1+µ + 1 r +µ1±γ y µ1, (2.2) x F 2 (x, y, t) L r Ct r +µ1 y µ1, (2.21) and therefore estimate (2.1) is proved. From estimates (2.7) and (2.1) we easily get result (2.3) of Lemma 2.1. To obtain (2.4) firstly we prove the following estimate G(x, y, t) = t 1/ sin π{} Λ (xt 1/ ) + y µ O(t 1+µ ) (2.22) for t, where x, y >. We write the representation (2.5) for the function F 1 (x, t) as F 1 (x y, t) = F 1 (x, t) + [F 1 (x y, t) F 1 (x, t)]. After the change of variables p t = z, we easily find that F 1 (x, t) = 1 2πi i e xp K(p)t dp = t 1/ 1 2πi i e zs K(z) dz. (2.23) Using the estimate e py 1 C py µ for y > and p (, i ) with µ 1 [, 1] we have F 1 (x y, t) F 1 (x, t) Therefore, from (2.23), we obtain i Cy µ t 1+µ F 1 (x y, t) = t 1/ 1 2πi e xp Cpt (e py 1) dp 1 i dz z µ e xt 1 z C z 2πi ). Cy µ t 1+µ = y µ O ( t 1+µ i (2.24) e zs K(z) dz + y µ O(t 1+µ ). (2.25) Now we write the representation (2.6) of the function F 2 (x, y, t) as F 2 (x, y, t) = M(x, y, t) + R(x, y, t), (2.26)

6 6 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 where M(x, t) = F 2 (x,, t) and R = [F 2 (x, y, t) F 2 (x,, t)]. Considering that e Cξ1/y 1 C ξ 1/ y µ and changing variables ξt = q and p t = z, we get Therefore, where F 2 (x, y, t) F 2 (x,, t) Ct 1 µ 1 y µ 1 i = y µ O(t 1+µ ). i dz e xt 1 z z 1 dq q 1 1+ µ e q K(z) + q F 2 (x, y, t τ) = M(x, t) + y µ O(t 1+µ ), (2.27) M(x, t) = t 1/ 1 i i 4π 2 dze sz K(z)z 1 dqe q q 1 κ(q) K(z) + q, (2.28) with s = xt 1/. Applying the Cauchy Theorem, we obtain i = 2πi i dze sz K(z)z 1 dqe q q 1 κ(q) K(z) + q i = I 1 + I 2, dze sz e K(z) + i dze sz K(z)z 1 where Γ = {z ( e iπ, e iπ ) (e iπ, e iπ )}. Using i dze sz dqe q 1 Γ K(z) + q =, we get M(x, t) = t 1/ 1 2πi i sin ( π{} ) i dze sz e Cz + t 1/ C( 1) {} 2π 2 i Γ dq e q q 1 κ(q) K(z) + q A 1 π A 1 dze sz z {} dqe q K(z) + q, q {} where A 1 = ( 1) {}+1 ( C ) [] Γ(1 {})Γ({}) sin π{}. From (2.25), (2.27), (2.3), we obtain (2.22) and then estimate (2.4). completes the proof of Lemma 2.1. We defined the space W = {φ(x, t) C((, ); L 1 L,µ ) : φ W < } where φ W = ( φ L 1 + φ L,µ). Lemma 2.2. The following two estimates are valid x µ t x t G(t τ)φ(τ)dτ L r C t r + µ t 1+γ φ W, G(t τ)φ(τ)dτ L r Ct r t 1 t 1+γ φ W, (2.29) (2.3) This for all t >, µ (, 1), r 1 and small enough γ >, provided that the right-hand sides are bounded.

7 EJDE-26/89 NONLINEAR PSEUDODIFFERENTIAL EQUATIONS 7 Proof. Let t [, 1]. By estimate (2.21) of Lemma 2.1, we get x F 2 (x, y, t) L r x L 1 y Ct r for all t >. Therefore, using (2.8) of Lemma 2.1 we obtain ỹ µ1 dy Ct r, (2.31) t x µ G(t τ)φ(τ)dτ L r t dτ( φ L r x µ F 1 L 1 + x µ φ L r F 1 L 1) + < C(t r +1 + t r + µ +1 ) φ W < C φ W t dτ φ L x µ F 2 L r x L 1 y (2.32) for all t [, 1]. Using (2.31) and (2.9), we have t x G(t τ)φ(τ)dτ L r t dτ( φ L r x F 1 L 1 + φ L x F 2 L r x L 1 y ) < Ct r +1 φ W t r φ W (2.33) for all t [, 1]. We consider now the case t > 1, t x µ G(t τ)φ(τ)dτ L r t/ t t/2 t/2 t t/2 dτ( φ L 1 x µ F 1 L r + x µ φ L 1 F 1 L r) dτ( φ L r x µ F 1 L 1 + x µ φ L r F 1 L 1) dτ φ L 1 x µ F 2 L r x L y dτ φ L x µ F 2 L r x L 1 y. So in view of estimates (2.2), (2.21), (2.8) and (2.9), we attain x µ t G(t τ)φ(τ)dτ L r Ct r + µ t 1+γ φ W. (2.34) In the same way we estimate x Gφ L r for t > 1, t x G(t τ)φ(τ)dτ L r t/2 + t t/2 dτ( φ L 1 x F 1 L r + φ L 1 x F 2 L r L ) dτ( φ L r x F 1 L 1 + φ L x F 2 L r x L 1 y ) Ct r t 1+γ φ W. Then, by (2.32)-(2.35) Lemma 2.2 is proved. (2.35)

8 8 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 Denote by Z = {φ(x) H 1, 2 H,µ H, 1 ; φ z < + }, φ Z = ( x µ φ L + φ L 1 + φ H 1 2 ), where µ (, 1]. Now we prove local existence theorem. Note that the existence time T > could be sufficiently small. Theorem 2.3. Let initial data u Z. Then for some time interval T > there exists a unique solution u C([, T ]; X) to problem (1.1). Moreover the existence time T can be chosen as follows ( ( 1 T = u Z 1 + 2C u ) ) 1 σ 1 µ 1 Z, µ1 = 1 1. Proof. We apply the contraction mapping principle in a ball of a radius ρ > in a complete metric space X T, X T,ρ = {φ C([, T ]; X) : sup t [,T ] u X = u XT where ρ = 1 2C u Z. For v X T,ρ we define the mapping M(v) by M(v) = G(t)u t where N (v(τ)) = λ v σ v. We first prove that M(v) XT ρ, ρ}, G(t τ)n (v(τ))dτ, (2.36) when v X T,ρ. We have by Lemmas 2.1 and 2.2 for µ 1 = 1 1, M(v) XT Gu XT + t G(t τ)n (v(τ))dτ XT C u Z + CT µ1 (1 + v XT ) σ+1 ρ 2 + CT µ1 (1 + ρ) σ+1 ρ, (2.37) if T > is small enough. Therefore, the mapping M transforms a ball of a radius ρ > into itself in the space X T. As in the proof of (2.37), we have for w, v X T M(w) M(v) XT t G(t τ)(n (w(τ)) N (v(τ)))dτ XT CT µ1 w v XT (1 + w XT + v XT ) σ 1 2 w v X T, since T > is small enough. Thus M is a contraction mapping in X T,ρ; therefore, there exists a unique solution u X T to the problem (1.1). Theorem 2.3 is proved. Now we define the space X[T 1, T 2 ] = C([T 1, T 2 ]; Z) with the norm ψ X[T1,T 2] = sup ψ(t) Z. t [T 1,T 2] Theorem 2.4. Let the initial data u Z and the following a priory estimate be valid u X[,T ) C(T ) u Z, (2.38) provided that there exists a solution u X[, T ) for some T >. Then there exists a unique global solution u X[, ) to the problem (1.1).

9 EJDE-26/89 NONLINEAR PSEUDODIFFERENTIAL EQUATIONS 9 Proof. From Lemma 2.1, we have that G(t T 1 ) : Z X[T 1, T 2 ] for any T 2 > T 1 and G(t T 1 )φ X[T1,T 2] C φ Z. Also using Lemma 2.2 we obtain that t T 1 G(t τ)n (v(τ))dτ X[T 1, T 2 ] for any v X[T 1, T 2 ], T 2 > T 1, and t G(t τ)(n (w(τ)) N (v(τ)))dτ X[T1,T 2] T 1 C w v X[T1,T 2](1 + w X[T1,T 2] + v X[T1,T 2]) σ, for all v, w X[T 1, T 2 ], where σ >. Using a priory estimates (2.38) we can prolongate the local solution given by Theorem 2.3 for all times t >. Indeed, by the contrary we can suppose that there exists a maximal existence time T > such that u X[, T ). If we choose a new initial time T 1 [, T ) and consider the problem (1.1) with initial data u(t 1 ), then via a priory estimate (2.38) the norm u(t 1 ) Z is bounded uniformly with respect to T 1 [, T ). Then the existence time given by the local existence Theorem 2.3 is bounded from below uniformly with respect to T 1 [, T ). Therefore if a new initial time T 1 > is chosen to be sufficiently close to the maximal time T, then by virtue of the local existence Theorem 2.3 we can guarantee that there exists a unique solution u X[, T ]. Now putting u(t ) as a new initial data at time T we can apply the local existence Theorem 2.3 and prolongate the solution u(t) on some bigger time interval [, T + T 2 ]. This contradicts to the fact that T is a maximal existence time. Hence there exists a unique solution u X[, ) to the problem (1.1). Theorem 2.4 is proved. 3. Large initial data (proof of Theorem 1.1) To prove of Theorem 1.1 we first apply the so-called energy method to estimate the L 2 (R) norm: i.e. we multiply equation (1.1) by u and integrate with respect to x R +, to get d dt u(t) 2 L 2 + 2λ By the Plancherel Theorem we have 2 uludx = 2 Re 1 2πi = Re(V P 1 πi i u σ+1 dx = 2 i Re u()v P 1 πi û(p)c p (û(p) u() p ) dp C p û(p) u() p i uludx. (3.1) 2 dp C p 1 (û(p) u() p ) dp). By the Cauchy Theorem using the analyticity of û(p) and K(p) = C p in the right-half complex plane we attain V P i C p 1 (û(p) u() p ) dp = 1 2 res p=(c p 1 (û(p) u() p )) + i C p 1 (û(p) u() ) dp =. p

10 1 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 Thus from dissipative condition Re K(p) > for Re p = we get 2 Also since λ > we have uludx = 1 π V P λ Therefore, from equation (3.1) we obtain Re K(ip) û(ip) u() ip 2 dp >. u σ+1 dx >. d dt u 2 L2. (3.2) Hence integrating with respect to time we see that sup u(t) L 2 u L 2. (3.3) t Now we prove that sup u x (t) L 2 u x L 2. (3.4) t We differentiate (1.1) with respect to the space variable, multiply equation (1.1) by u x and integrate with respect to x R +, to get Since and 2 d dt u x(t) 2 L 2 + 2σλ u x x Ludx = 1 π V P σλ u σ 1 u 2 xdx = 2 u x x Ludx. (3.5) Re K(ip) p 2 û(ip) u() ip 2 dp > u σ 1 u 2 xdx > integrating (3.5) with respect to time we easily get (3.4). Note that time decay estimates (3.3) and (3.4) are not optimal. To get an optimal time decay estimates we need to show that the L 1 - norm of the solution does not grow with time. Using the idea of papers [2], [5] we multiply equation (1.1) by S sign u u u and integrate with respect to x over R+ to get u t (x, t)s(x, t)dx + N (u)(x, t)s(x, t)dx = S(x, t)ludx, (3.6) R + R + R + where N (u) = λ u σ u. We have u t (x, t)s(x, t)dx = R + R t u(x, t) dx = d dt u(t) L 1, + N (u)(x, t)s(x, t)dx = λ u σ dx. R + R + Representing the operator Lu via the Riesz potential (see [29]) let us show that S(x, t)ludx. (3.7) R + By [1], we have 1 i p ν 1 e px 1 dp = 2πi Γ(ν + 1) xν.

11 EJDE-26/89 NONLINEAR PSEUDODIFFERENTIAL EQUATIONS 11 Thus for (1, 2), we get x Lu = C x (x y) 1 y u(y) dy, C >. Denote S(x, t) = sign(u(x, t)) and represent u(x, t) = S(x, t) u(x, t). We make a regularization { K ε 2 (x) = xx 1, for x ε, for x < ε, such that K ε(x) and K ε (x) for all x >. We can easily see that x x x (x y) 1 u y (y, t) dy = lim x K ε (x y)u y (y, t) dy. ε (To justify our calculation we note that the linear operator L in equation (1.1) is strongly dissipative, therefore by smoothing effect the solution obtain regularity u C 1 (R + ) (see Theorem 2.3 and [28])).We have x dxs(x, t) x K ε (x y) y u(y, t) dy R + x = dxs(x, t) x dyk ε (x y)s(y, t) y u(y, t) R + = K ε () R + dxs 2 (x, t) x u(x, t) + dy y u(y, t) R + y dxs(y, t)s(x, t) x K ε (x y). Then via the identity S(y, t)s(x, t) = (S(x, t) S(y, t))2, x dxs(x, t) x K ε (x y) y u(y, t) dy R + = K ε () dxs 2 (x, t) x u(x, t) + dy y u(y, t) dx x K ε (x y) R + R + y 1 dy y u(y, t) dx(s(x, t) S(y, t)) 2 x K ε (x y). 2 R + y Since R + dy y u(y, t) y dx x K ε (x y) = K ε () dy y u(y, t) R + we obtain x dxs(x, t) x K ε (x y) y u(y, t) dy R + = 1 dy y u(y, t) dx(s(x, t) S(y, t)) 2 x K ε (x y). 2 R + y (3.8)

12 12 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 Integrating by parts, we have R + dy y u(y, t) = u(, t) y + dy u(y, t) R + dx(s(x, t) S(y, t)) 2 x K ε (x y) dx(s(x, t) S(, t)) 2 x K ε (x) y dxk ε (x y)(s(x, t) S(y, t)) 2. Therefore, from (3.8) using x K ε (x) <, we gain x dxs(x, t) x K ε (x y) y u(y, t) dy R + 1 dy u(y, t) dxk ε (x y)(s(x, t) S(y, t)) 2 2 R + y and therefore since xk(x) 2 > for all x > we get x dxs(x, t) x K ε (x y) y u(y, t) dy. R + Hence we have S(x, t)ludx R + C 2 lim dy u(y, t) dxk ε (x y)(s(x, t) S(y, t)) 2. ε R + R + Thus (3.7) is true and from (3.6) we find d dt u(t) L 1. From this inequality, we see that the norm u(t) L 1 is bounded for all t. We now prove that the norm u x (t) L 2 as t. Taking ϱ (, 1) by the Plancherel theorem, u x x Ludx = 1 π V P where we have used that p ϱ Re K(ip) p 2 û(ip, t) u() ip 2 dp Re C p û x (p, t) 2 dp Cϱ u x (t) 2 L 2 Cϱ+3 sup û(p, t) 2 p <ϱ Cρ +2 u() sup û(p, t) Cϱ +1 u() 2 p <ϱ Cϱ u(t) 2 L 2 Cϱ+3 u(t) 2 L 1 Cϱ +2 u(t) L 1 u L Cϱ +1 u 2 L Cϱ u(t) 2 L 2 Cϱ+3 u(t) 2 L 1 Cϱ +2 u(t) L 1 u 1 2 L 2 u x 1 2 L 2 Cϱ +1 u L 2 u x L 2, u x (t) 2 L C u(t) L 2 u x(t) L 2. (3.9)

13 EJDE-26/89 NONLINEAR PSEUDODIFFERENTIAL EQUATIONS 13 Since the norms u(t) L 1, u(t) L 2 and u x (t) L 2 are bounded, choosing ϱ(t) = C 1/ (1 + t) 1/, we obtain d dt u x(t) 2 L 2 (1 + t) 1 u x (t) 2 L 2 + C(1 + t) 1 1. We substitute u x (t) 2 L = h(t)(1 + t) 2, then for h(t) we have h (t) C(1 + t) 1 1, hence integration with respect to time yields h(t) C(1 + t) 2 1. Therefore we get the time decay estimate and therefore u x (t) L 2 C(1 + t) 1/(2). u x (t) L C u(t) 1 2 L 2 u x (t) 1 2 L 2 C(1 + t) 1 4. We substitute this estimate in (3.9) to obtain d dt u x(t) 2 L 2 (1 + t) 1 u x (t) 2 L 2 + C(1 + t) Again after the change u x (t) 2 L = h(t)(1 + t) 2 we get u x (t) L 2 C(1 + t) We can repeat this consideration to get the optimal time decay estimate u x (t) L 2 C(1 + t) 3 2 (3.1) for all t >. We now prove that the norm u(t) L 2 as t. Taking ϱ (, 1) by the Plancherel theorem we get uludx = 1 i π V P Re K(ip)(û(ip, t) u() )û dp ip i Re C p û(p, t) 2 dp u() Re C p 1 û dp p ϱ ϱ u(t) 2 L 2 ϱ+1 sup û(p, t) 2 u() ϱ sup û(p, t) ϱ 1 u() 2 p <ϱ p <ϱ ϱ u(t) 2 L 2 ϱ+1 u(t) 2 L 1 u() ϱ u(t) L 1 ϱ 1 u() 2. Since the norms u(t) L 1, u(t) L 2 are bounded and due to (3.1) u() u(t) L C u(t) 1 2 L 2 u x (t) 1 2 L 2 C(1 + t) 3 4, choosing ϱ(t) = C 1/ (1 + t) 1/, we obtain d dt u(t) 2 L (1 + 2 t) 1 u(t) 2 1 L2 + C(1 + t) 1 2. We substitute u(t) 2 L = h(t)(1 + t) 2, then for h(t) we have 2 h (t) C(1 + t) 1 1 2,

14 14 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 hence integration with respect to time yields h(t) C(1 + t) Therefore, we have the time decay estimates We can repeat this consideration to get u(t) L 2 C(1 + t) 1 4, u() C(1 + t) d dt u(t) 2 L 2 (1 + t) 1 u(t) 2 L 2 + C(1 + t) 1 1. Therefore, we obtain the optimal estimate u(t) L 2 C(1 + t) 1 2 (3.11) for all t >. Also from (3.1) and (3.11) we get the optimal time decay of the L (R + )-norm of the solutions u(t) L C(1 + t) 1/ (3.12) for all t >. Now we can estimate the norm L,µ (R + ). By the integral formula (2.2) we have u(t) L,µ u L 1 G(t) L,µ + u L,µ G(t) L 1 + C + C Hence using Lemmas 2.1 and 2.2, u(t) L,+1 C t 1 µ C t 1 µ t t u(τ) σ+1 L 1 G(t τ) L,µdτ u(τ) σ+1 L,µ G(t τ) L 1dτ t t + C u(τ) σ 1 µ L t τ dτ + C u(τ) σ L u(τ) L,µdτ t + C u(τ) L,µ τ σ dτ. Hence for the function h(t) = sup τ t u(τ) L,µ, we get the inequality h(t) C t 1 µ t + C τ σ h(τ)dτ and since σ > by the Gronwall s lemma it follows that u(t) L,µ C t 1 µ t >. (3.13) From a priory estimates, due to Theorem 2.4 there exists a unique solution to the problem (1.1), such that u(x, t) C([, ); H 1, 2 H,µ H, 1 ) u(t) L C t 1/, u(t) L,µ C t 1 µ and u(t) L 2 C t 1 2. (3.14)

15 EJDE-26/89 NONLINEAR PSEUDODIFFERENTIAL EQUATIONS 15 By Lemma 2.2 we have sup t 1 (1+µ) Gφ t 1 Λ (xt 1/ )f(φ) L C φ L 1,µ, t>1 where µ [, 1]. Substituting this formula into (2.2) we obtain where by (3.14), and A = u(x, t) = t 1/ AΛ ( x ) + R(x, t), (3.15) t 1 u dy + dτ ( R(x, t) = O(t 1+µ ) y µ u (y) dy + + = O(t 1+µ ). ( τ 2µ O (t τ) 1+µ N (u) dy <, dτ ) + dτ N (u) dy Thus the asymptotic (1.4) is valid. Theorem 1.1 is then proved. References ) y µ N (u) dy [1] H. Bateman and A. Erdelyi; Tables of Integral Transforms, McGraw-Hill Book Co., N.Y., 1954, 343 pp. [2] C. Bardos, P. Penel, U. Frisch and P.L. Sulem; Modified dissipatively for a nonlinear evolution equation arising in turbulence, Arch. Rat. Mech. Anal. 71 (1979), pp [3] C. J. Amick, J. L. Bona and M. E. Schonbek; Decay of solutions of some nonlinear wave equations, J. Diff. Eqs. 81 (1989), pp [4] P. Biler, Asymptotic behavior in time of solutions to some equations generalizing the Korteweg-de Vries-Burgers equation, Bull. Polish Acad. Sci. Math. 32 (1984), No. 5-6, pp [5] P. Biler, T. Funaki and W. A. Woyczynski; Fractal Burgers equations, J. Diff. Eq. 148 (1998), pp [6] P. Biler, G. Karch and W. Woyczynski; Multifractal and Levy conservation laws, C.R. Acad. Sci., Paris, Ser.1, Math. 33 (2), no 5, pp [7] J. L. Bona, F. Demengel and K. S. Promislow; Fourier splitting and dissipation nonlinear dispersive waves, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), no. 3, pp [8] J. L. Bona and L. Luo, Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation, Advances in Nonlinear Mathematics and Science, Phys. D 152/153 (21), pp [9] D. B. Dix, Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation, J. Diff. Eqs. 9 (1991), pp [1] D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. P. D. E. 17 (1992), pp [11] H. Brezis and F. Browder; Partial differential equations in the 2th century, Adv. Math. 135 (1998), no. 1, pp [12] M. Escobedo, O. Kavian and H. Matano; Large time behavior of solutions of a dissipative nonlinear heat equation, Comm. Partial Diff. Eqs., 2 (1995), pp [13] M. Escobedo, J. L. Vázquez and E. Zuazua; Asymptotic behavior and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124 (1993), pp [14] Th. Cazenave and F. Dickstein; On the influence of boundary conditions on flows in porous media, J. Math. Anal. Appl. 253 (21), no. 1, pp [15] V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskii; On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation, Math. USSR Sbornik 54 (1986), pp

16 16 R. E. CARDIEL, E. I. KAIKINA EJDE-26/89 [16] Y. Giga and T. Kambe; Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation, Commun. Math. Phys. 117 (1988), pp [17] A. Gmira and L. Veron; Large time behavior of the solutions of a semilinear parabolic equation in R n, J. Diff. Eqs. 53 (1984), pp [18] N. Hayashi, E. I. Kaikina and P. I. Naumkin; Large time behavior of solutions to the Landau- Ginzburg type equations, Funkcialaj Ekvacioj, 44 (1) (21), pp [19] N. Hayashi, E. I. Kaikina and P. I. Naumkin; Large time behavior of solutions to the dissipative nonlinear Schrödinger equation, Proceedings of the Royal Soc. Edingburgh, 13-A (2), pp [2] N. Hayashi and E. I. Kaikina; Nonlinear Theory of Pseudodifferential Equations on a Half- Line, Mathematics Studies, Elsevier, 194 (23). [21] G. Karch, Long-time behavior of solutions to non-linear wave equations: higher-order asymptotics. Math. Methods in Appl. Sci. 22 (1999), no.18, pp [22] O. Kavian, Remarks on the large time behavior of a nonlinear diffusion equation, Ann. Inst. Henri Poincaré, Analyse non linéaire, 4, No. 5 (1987), pp [23] E. I. Kaikina, Nonlinear Pseudodifferential Equations on a Segment, Diff. Int. Eq., 18 (25), no. 2, pp [24] E. I. Kaikina, Nonlinear nonlocal Ott-Sudan-Ostrovskiy type equations on a segment, Hokkaido Univ. J., to appear. [25] E. I. Kaikina, P. I. Naumkin and I. Sánchez-Suárez; Nonlinear nonlocal Schrödinger type equations on a segment, SUT J. Math., 4 (24), no. 1, pp [26] P. I. Naumkin, I. A. Shishmarev, Asymptotics as t of solutions of generalized Kolmogorov-Petrovskii-Piskounov equation, Mat. Modelirovanie 1, No. 3 (1989), pp (Russian). [27] P. I. Naumkin, I. A. Shishmarev, Asymptotics of solutions of the Whitham equation for large time, Mat. Modelirovanie, 2, No. 3 (199), pp (Russian). [28] P. I. Naumkin and I. A. Shishmarev; Nonlinear Nonlocal Equations in the Theory of Waves, Transl. Math. Monographs, AMS, Providence, R.I., 133, [29] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, 3, Princeton Univ. Press, Princeton, NJ, 197. [3] M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal. T.M.A. 1 (1986), no. 9, pp [31] M. E. Schonbek, Lower bounds of rates of decay for solutions to the Navier-Stokes equations, J. Amer. Math. Soc. 4 (1991), no.3, pp [32] W. A. Strauss, Nonlinear scattering at low energy, J. Funct. Anal. 41 (1981), pp [33] Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping; The critical case, C.R. Acad. Sci. Paris, Série I, 333 (21), pp [34] E. Zuazua, Weakly non-linear large time behavior for scalar convection-diffusion equations, Diff. Int. Eqs. 6, No. 5 (1993), pp [35] E. Zuazua, A dynamical system approach to the self-similar large time behavior in scalar convection-diffusion equations, J. Diff. Eqs. 18 (1994), pp Rosa E. Cardiel Instituto de Matemáticas UNAM (Campus Cuernavaca), Av. Universidad s/n, col. Lomas de Chamilpa, Cuernavaca, Morelos, Mexico address: rosy@matcuer.unam.mx Elena I. Kaikina Instituto de Matemáticas, UNAM Campus Morelia, AP 61-3 (Xangari), Morelia CP 5889, Michoacán, Mexico address: ekaikina@matmor.unam.mx