Related papers: Exponential decay of semigroups for second order n…
In $L_2 (\mathbb{R}^d; \mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$ with periodic coefficients depending on $\mathbf{x}/\varepsilon$. We find approximations…
For a class of weakly hyperbolic systems of the form D_t - A(t,x,D_x), where A(t,x,D_x) is a first-order pseudodifferential operator whose principal symbol degenerates like t^{l_*} at time t=0, for some integer l_* \geq 1, well-posedness of…
For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann-Hilbert…
Domain decomposition methods are essential in solving applied problems on parallel computer systems. For boundary value problems for evolutionary equations the implicit schemes are in common use to solve problems at a new time level…
Parabolic integro-differential Kolmogorov equations with different space-dependent operators are considered in H\"{o}lder-type spaces defined by a scalable L\'{e}vy measure. Probabilistic representations are used to prove continuity of the…
The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=\Div B^*\nabla(a\Div A\nabla u)=0, where A and B are elliptic matrices with complex-valued bounded…
In this paper we deal with the Cauchy problem for the incompressible Euler equations in the three-dimensional periodic setting. We prove non-uniqueness for an $L^2$-dense set of H\"older continuous initial data in the class of H\"older…
In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equation with Caputo fractional derivative in time. The second critical exponent appears if the…
The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported.
This paper is concerned with solution in H\"{o}lder spaces of the Cauchy problem for linear and semi-linear backward stochastic partial differential equations (BSPDEs) of super-parabolic type. The pair of unknown variables are viewed as…
The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order {\alpha}\in(0,2) is investigated. The principal part of the operator has kernel…
We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \partial_t u -\Delta_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where $N\ge 1$,…
We study heat and wave type equations on a separable Hilbert space $\mathcal{H}$ by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the…
Let $L$ be an unbounded linear operator in a real Hilbert space $H$, a generator of $C_0$ semigroup, and $g:H\to H$ be a $C^2_{loc}$ nonlinear map. The DSM (dynamical systems method) for solving equ$ $F(v):=Lv+gv=0$ consists of solving the…
We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u +…
We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schr\"odinger (NNLS) equation $ iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $ with the step-like initial data close to the ``shifted step function''…
We study the Cauchy problem for the defocusing nonlinear Schr\"odinger (NLS) equation under the assumption that the solution vanishes as $x \to + \infty$ and approaches an oscillatory plane wave as $x \to -\infty$. We first develop an…
We study Cauchy problems of fractional differential equations in both space and time variables by expressing the solution in terms of ``stochastic composition" of the solutions to two simpler problems. These Cauchy sub-problems respectively…
We study the splitting scheme associated with the linear stochastic Cauchy problem dU(t) = AU(t) dt + dW(t), where A is the generator of an analytic C_0-semigroup S={S(t)} on a Banach space E and W={W(t)} is a Brownian motion with values in…
We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is…