The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

and Applied Analysis 3 use the Kato theorem 29 to establish the local well-posedness for 1.5 with initial value u0 ∈ H with s > 3/2. Then, we present a precise blow-up scenario for 1.5 . Provided that u0 ∈ H R ⋂ L1 R and the potential y0 1−∂x u0 does not change sign, the global existence of the strong solution is shown to be true. Finally, under suitable assumptions, the existence and uniqueness of global weak solution in W1,∞ R × R ⋂ Lloc R ;H 1 R are proved. Our main ideas to prove the existence and uniqueness of the global weak solution come from those presented in Constantin and Molinet 8 and Yin 22 . 2. Notations The space of all infinitely differentiable functions φ t, x with compact support in 0, ∞ ×R is denoted by C∞ 0 . Let 1 ≤ p < ∞, and let L L R be the space of all measurable functions h t, x such that ‖h‖PLP ∫ R |h t, x |dx < ∞. We define L∞ L∞ R with the standard norm ‖h‖L∞ infm e 0supx∈R\e|h t, x |. For any real number s, let H H R denote the Sobolev space with the norm defined by ‖h‖Hs ∫ R 1 |ξ| |ĥ t, ξ |dξ 1/2 < ∞, where ĥ t, ξ ∫ R e −ixξh t, x dx. We denote by ∗ the convolution. Let ‖ · ‖X denote the norm of Banach space X and 〈·, ·〉 the H1 R , H−1 R duality bracket. Let M R be the space of the Radon measures on R with bounded total variation and M R the subset of positive measures. Finally, we write BV R for the space of functions with bounded variation, V f being the total variation of f ∈ BV R . Note that if G x : 1/2 e−|x|, x ∈ R. Then, 1 − ∂x −1f G ∗ f for all f ∈ L2 R and G ∗ u − uxx u. Using this identity, we rewrite problem 1.5 in the form ut buux ∂xG ∗ [ a 2 u2 3b − a 2 ux 2 ] λu 0, t > 0, x ∈ R, u 0, x u0 x , x ∈ R, 2.1 which is equivalent to yt buyx ayux λy 0, t > 0, x ∈ R, y u − uxx, u 0, x u0 x . 2.2 3. Preliminaries Throughout this paper, let {ρn}n≥1 denote the mollifiers ρn x : (∫ R ρ ξ dξ )−1 nρ nx , x ∈ R, n ≥ 1, 3.1 4 Abstract and Applied Analysis where ρ ∈ C∞ c R is defined by ρ x : ⎧ ⎨ ⎩ e1/ x 2−1 for |x| < 1, 0 for |x| ≥ 1. 3.2