We study large time asymptotics of small solutions to the Cauchy problem for the one dimensional nonlinear damped wave equationin the sub critical case ∈ 2 − ε 3 , 2 . We assume that the initial data v 0 , 1 + * xAlso we suppose that the mean value of initial dataThen there exists a positive value ε such that the Cauchy problem (1) has a unique global solution v (t, x) ∈ C [0, ∞) ; L ∞ ∩ L 1,a , satisfying the following time decay estimate:v (t) L ∞ Cε t − 1 for large t > 0, here 2 − ε 3 < < 2.

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Abstract. We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearitywhere ε > 0, and space dimensions n = 1, 2, 3. Assume that the initial data, weighted Sobolev spaces areThen we prove that there exists a positive ε 0 such that the Cauchy problem above has a unique global solution u ∈ C [0, ∞) ; H δ,0 satisfying the time decay property

We study the Cauchy problem for the nonlinear Schr odinger equation with dissipation ut + L u + ijuj 2 u = 0;x 2 R; t > 0;where L is a linear pseudodi® erential operator with dissipative symbol Re L(¹ ) > C1 j¹ j 2 =(1 + ¹ 2 ) and jL 0 (¹ )j 6 C2 (j¹ j + j¹ j n ) for all ¹ 2 R. Here, C1 ; C2 > 0, n > 1. Moreover, we assume that L(¹ ) = ¬ ¹ 2 + O(j¹ j 2+ ® ) for all j¹ j < 1, where ® > 0, Re ¬ > 0, Im ¬ > 0. When L(¹ ) = ¬ ¹ 2 , equation (A) is the nonlinear Schr odinger equation with dissipation ut ¬ uxx + ijuj 2 u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimateunder the conditions that u0 2 H n;0 \ H 0;1 have the mean valuêand the norm ku0 k H n;0 + ku0 k H 0;1 = " is su± ciently small, where ¼ = 1 if Im ¬ > 0 and ¼ = 2 if Im ¬ = 0, and H m ;s = f¿ 2 S 0 ; k¿ km ;s = k(1 + x 2 ) s=2 (1 @ 2 x ) m =2 ¿ k < 1 g; m; s 2 R:Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation ut ¬ uxx + ijuj p 1 u = 0, with p > 3 have the same time decay estimate kuk L 1 = O(t 1=2 ) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

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