Nonlinear nonlocal Whitham equation on a segment
The author studies global existence and asymptotic behavior of solutions to the initial-boundary value problem for the nonlinear nonlocal Whitham equation on a segment \[ \begin{cases} u_t+uu_x+{K}u=0, &t>0,\;x\in(0,a),\\ u(x,0)=u_0(x), &x\in(0,a),\\ u(a,t)=0, &t>0, \end{cases} \tag{1} \] where the pseudodifferential operator \({K}u\) is defined on a segment \([0,a]\) by \[ {K}u= \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}e^{px}K(p) \times\left(\widehat{u}(p,t)-\frac{u(0,t)-e^{-pa}\,u(a,t)}{p}\right)\,dp, \] where \(K(p)=C_\alpha p^\alpha\), \(\alpha\in(1,3/2)\) and \(C_\alpha\) is chosen such that \(\text{Re}\,K(p)>0\) for \(\text{Re}\,p=0\). She proves that, for initial data \(u_0\in {L}^\infty\) and \(\| u_0\| _{L^\infty}<\varepsilon\), the initial-boundary value problem ({1}) has a unique solution in \({C}([0,\infty);{L}^2(0,a))\). The solution has the following large time asymptotics \[ u(x,t)=At^{-1/\alpha}\Lambda+O(t^{-(1+\delta)/\alpha}), \] for some \(A\) and uniformly with respect to the spatial variable \(x\in(0,a)\), where \[ \Lambda=\frac{e^{-i\pi\alpha/2}\cos\pi\alpha/2}{\pi i} \int_0^{+i\infty} e^{-K(z)}\,dz. \]
dissipative nonlinear evolution equation
scientific article; zbMATH DE number 2117562