Nonlinear nonlocal Whitham equation on a segment Original Research Article

We study global existence and large time asymptotic behavior of solutions to the initial-boundary value problem for the nonlinear nonlocal equation on a segment equation(1) View the MathML sourceut+uux+Ku=0,t>0,x∈(0,a),u(x,0)=u0(x),x∈(0,a),u(a,t)=0,t>0, Turn MathJax on where the pseudodifferential operator KuKu on a segment [0,a][0,a] is defined by View the MathML sourceKu=12πi∫-i∞i∞epxK(p)×u^(p,t)-u(0,t)-e-pau(a,t)pdp, Turn MathJax on with a symbol K(p)=CαpαK(p)=Cαpα, View the MathML sourceα∈1,32, CαCα is chosen such that ReK(p)>0ReK(p)>0 for Rep=0Rep=0. We prove that if the initial data u0∈L∞u0∈L∞ and ∥u0∥L∞<ɛ∥u0∥L∞<ɛ, then there exists a unique solution u∈C([0,∞);L2(0,a))u∈C([0,∞);L2(0,a)) of the initial-boundary value problem (0.1). Moreover, there exists a constant A such that the solution has the following large time asymptotics u(x,t)=At-1/αΛ+O(t-(1+δ)/α)),u(x,t)=At-1/αΛ+O(t-(1+δ)/α)), Turn MathJax on uniformly with respect to the spatial variable x∈(0,a)x∈(0,a), where View the MathML source