Nonlinear pseudodifferential equations on a segment.(English)Zbl 1212.35417

Summary: We study the global existence and large-time asymptotic behavior of solutions to the initial/boundary-value problem for the nonlinear nonlocal Whitham equation on a segment $$(0,a)$$, $\begin{cases} u_t+uu_x+\mathbb Ku=0,\quad t>0,\,x\in (0,a) \\ u(x,0)=u_0(x),\quad x\in (0,a), \end{cases}$ where the pseudodifferential operator $$\mathbb Ku$$ on a segment $$[0,a]$$ is defined by $\mathbb Ku=\theta _a(x)\frac {1}{2\pi \text{i}} \int _{-\text{i}\infty }^{\text{i}\infty } \text{e}^{px}K(p)\Big (\widehat u(p,t)-\frac {u(0,t)-\text{e}^{-pa}u(a,t)}{p}\Big)\,\text dp, \tag{1}$ where $$K(p)=C_{\alpha }p^{\alpha }$$, $$\alpha \in (\frac {3}{2},2)$$, and $$C_{\alpha }$$ is chosen by the dissipation conditions. We prove that if the initial data $$u_0=\mathbf L^{\infty }(0,a)$$ have a small norm $$\| u_0\| _{\mathbf {L}^{\infty }}<\varepsilon$$, then there exists a unique solution $$u\in \mathbf {C}([0,\infty);\mathbf {L}^2(0,a)) \cap \mathbf {C}((0,\infty);\mathbf {H}^1 (0,a))$$ to problem $$(1)$$. Moreover, there exists a function $$A(x)\in \mathbf {L}^{\infty }(0,a)$$ such that the solution has the following asymptotics for large time $$t\to \infty$$: $u(x,t)=A(x)Bt^{-\frac {1}{\alpha }}+O(t^{-\frac {1+\delta }{\alpha }}),$ uniformly with respect to $$x\in (0,a)$$, where $$\delta \in (0,2-\alpha)$$.

MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35S15 Boundary value problems for PDEs with pseudodifferential operators 35B40 Asymptotic behavior of solutions to PDEs

Whitham equation