We prove well-posedness in $L^2$-based Sobolev spaces $H^s$ at high regularity for a class of nonlinear higher-order dispersive equations generalizing the KdV hierarchy both on the line and on the torus.

Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem $$ \mathbb{D}_t^\alpha u(t)=Au(t)+f(t), \;t > 0,\;\;0 < \alpha\le 1, $$ where $\mathbb D_t^\alpha$ is the Caputo fractional derivative of order $\alpha$, $A$ a closed linear operator on some Banach space $X$, $f:\;[0,\infty)\to X$ is a given function. We define an operator family associated with this problem and study its regularity properties. When $A$ is the generator of a $\beta$-times integrated semigroup $(T_\beta(t))$ on a Banach space $X$, explicit representations of mild and classical solutions of the above problem in terms of the integrated semigroup are derived. The results are applied to the fractional diffusion equation with non-homogeneous, Dirichlet, Neumann and Robin boundary conditions and to the time fractional order Schrödinger equation $\mathbb{D}_t^\alpha u(t,x)=e^{i\theta}\Delta_pu(t,x)+f(t,x),$ $ t > 0,\; x\in \mathbb R ^N$ where $\pi/2\le \theta < (1-\alpha/2)\pi$ and $\Delta_p$ is a realization of the Laplace operator on $L^p(\mathbb R ^N)$, $1\le p < \infty$.

The spectral problem of anisotropic elasticity with traction-free boundary conditions is considered in a bounded domain with a spatial cusp having its vertex at the origin and given by triples $(x_1,x_2,x_3)$ such that $x_3^{-2}(x_1,x_2) \in \omega$, where $\omega$ is a two-dimensional Lipschitz domain with a compact closure. We show that there exists a threshold $\lambda_\dagger>0$ expressed explicitly in terms of the elasticity constants and the area of $\omega$ such that the continuous spectrum coincides with the half-line $[\lambda_\dagger,\infty)$, whereas the interval $[0,\lambda_\dagger)$ contains only the discrete spectrum. The asymptotic formula for solutions to this spectral problem near cusp's vertex is also derived. A principle feature of this asymptotic formula is the dependence of the leading term on the spectral parameter.

In this paper, we present a long time existence theory for a new enhanced Boussinesq-Type system with constant bathymetry written in a conservative form. We also prove the existence of solitary wave for a large class of asymptotic models, including Beji-Nadaoka, Madsen-Sorensen and Nwogu equations. Furthermore, we give a procedure to calculate numerically these particular solutions and we present some effective computations.

In this paper, we consider the initial boundary value problem (IBVP) of the Hamiltonian fifth-order KdV equation posed on a finite interval $(0,L)$, $$ \begin{cases} \partial_t u-\partial_x^{5}u= c_1 u\partial_x u+ c_2 u^2 \partial _x u + 2b\partial_x u\partial_x^{2}u+bu\partial_x^{3} u, \quad x\in (0,L), \ t>0 \\ u(0,x)=\phi (x) , \ x\in (0,L)\\ u(t,0)=\partial_x u(t,0)=u(t,L)=\partial_x u(t,L)=\partial_x^{2}u(t,L)=0, \quad t>0, \end{cases} $$ and show that, given $0\leq s\leq 5$ and $T>0$, for any $\phi \in H^s (0,L) $ satisfying the natural compatibility conditions, the IBVP admits a unique solution $$ u\in L^{\infty}_{loc} (\mathbb R^+; H^s(0,L))\cap L^2 _{loc}(\mathbb R^+; H^{s+2} (0,L)). $$ Moreover, the corresponding solution map is shown to be locally Lipschtiz continuous from $L^2 (0,L)$ to $L^{\infty}(0,T; L^2 (0,L))\cap L^2 (0,T; H^2 (0,L))$ and from $H^5 (0,L)$ to $L^{\infty}(0,T; H^5 (0,L))\cap L^2 (0,T; H^7 (0,L))$, respectively, for any given $T>0$ . This is in sharp contrast to the pure initial value problem (IVP) of the equation posed on the whole line $\mathbb R$, $$ \begin{cases} \partial_t v-\partial_x^{5}v= c_1 v\partial_x v+ c_2 v^2 \partial _x v + 2b\partial_x v\partial_x^{2}u+bv\partial_x^{3} v, \quad x\in \mathbb R, \ t\in \mathbb R \\ v(0,x)=\psi (x) , \ x\in \mathbb R, \end{cases} $$ which is known to be (globally) well-posed in the space $H^s (\mathbb R)$ for $s\geq 2$ and the corresponding solution map is continuous, but fails to be uniformly continuous on any ball in $H^s(\mathbb R)$.