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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
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