We prove the local Hadamard well-posedness of the higher-order nonlinear Schr\"odinger equation with negative dispersion coefficient on the half-line under nonzero Dirichlet and Neumann boundary data. The paper uncovers a striking feature of this model: unlike the positive-dispersion case studied in (Alk{\i}n, Mantzavinos, Ozsari, 2024), well-posedness requires two boundary conditions rather than one. This reflects a structural limitation in the underlying global relation, which allows elimination of at most one unknown boundary trace. Using the unified transform of Fokas, we derive an explicit solution formula for the associated forced linear problem and establish sharp Sobolev and Strichartz estimates on the half-line. These linear estimates, together with corresponding Cauchy problem bounds, yield local well-posedness in both high- and low-regularity Sobolev spaces, as well as local Lipschitz continuity of the solution map.
We prove local well-posedness for the initial-boundary-value problem for the Korteweg--de Vries equation on the right half-line at the critical regularity $s=-3/4$, with initial and boundary data in $H^{-3/4}(\mathbb R_+)$ and $H^{1/12}(\mathbb R_+)$, respectively. This reaches the endpoint left open in the previous ha...
We study the Kuramoto--Sivashinsky (KS) equation in one, two, and three spatial dimensions subject to a codimension-one constraint. The constraint preserves the \(L^2\)-norm of the solution and consequently provides a uniform a priori bound that plays a central role in the higher-order energy estimates. In one spatial...
We establish the global well-posedness for the energy-subcritical defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on the three-dimensional Friedlander model domain \( \Omega=\{x\in\mathbb{R}:x\geq0\}\times\mathbb{R}^2_{y,z}, \) subject to homogeneous Dirichlet boundary conditions, for arbitrarily large initia...
We study the initial–boundary value problem for a fractional nonlinear Klein–Gordon equation posed on the hyperoctant Dn,n⩾2 with nonhomogeneous boundary data on each coordinate hyperplane. The equation involves a spatial fractional Laplacian of Caputo type with order β∈(3/2,2), which induces strong nonlocal dissipatio...
This paper establishes global well-posedness for the defocusing energy-critical nonlinear Schr\"odinger equation on the three-dimensional torus with a class of additive L\'evy noises. For arbitrary deterministic initial data in the energy space, we construct probabilistically strong solutions that are pathwise unique i...
Dispersive and Strichartz estimates are fundamental tools for establishing the well-posedness and long-time behavior of solutions to nonlinear partial differential equations. While these estimates are well-understood in the boundaryless Euclidean setting, the presence of a geometric boundary introduces severe analytica...
Len Meas· 0 citations
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