Related papers: On the Classifications of Scalar Evolution Equatio…
For the problem $$ \left\{ \begin{aligned} & \partial_t^k u - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) \ge f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty), & u (x, 0) = u_0 (x), \: \partial_t u…
This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation…
In the present work, a new time-dependent exchange theory is presented wherein the symmetry constraints, on a multi-electron wavefunction, are properly accounted for. In so doing, the equations of motion, incorporating the required…
We investigate the dispersive properties of evolution equations on waveguides with a non flat shape. More precisely we consider an operator $H=-\Delta_{x}-\Delta_{y}+V(x,y)$ with Dirichled boundary condition on an unbounded domain $\Omega$,…
We consider the optimization of a dynamical system by switching at discrete time points between abstract evolution equations composed by nonlinearly perturbed strongly continuous semigroups, nonlinear state reset maps at mode transition…
We consider semilinear higher order (in time) evolution inequalities posed in an exterior domain of the half-space $\mathbb{R}_+^N$, $N\geq 2$, and involving differential operators of the form $\mathcal{L}_\lambda =-\Delta +\lambda/|x|^2$,…
In the first part we present a generalized implicit function theorem for abstract equations of the type $F(\lambda,u)=0$. We suppose that $u_0$ is a solution for $\lambda=0$ and that $F(\lambda,\cdot)$ is smooth for all $\lambda$, but,…
We look at estimates for the Green's function of time-fractional evolution equations of the form $D^{\nu}_{0+*} u = Lu$, where $D^{\nu}_{0+*}$ is a Caputo-type time-fractional derivative, depending on a L\'evy kernel $\nu$ with variable…
This paper deals with differential equations of the form $$ \tau(y)- \lambda ^{2m} \varrho(x) y = 0, \quad \tau(y) =\sum_{k,\,s=0}^m(\tau_{k,\,s}(x)y^{(m-k)}(x))^{(m-s)}, $$ where $n=2m\geqslant 2$, $\lambda$ is the large complex parameter,…
In this paper, we study the following class of fractional Hamiltonian systems: \begin{eqnarray*} \begin{aligned}\displaystyle \left\{ \arraycolsep=1.5pt \begin{array}{ll} (-\Delta)^{\frac{1}{2}} u + u = \Big(I_{\mu_{1}}\ast G(v)\Big)g(v) \…
In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,x~n = F^(n)(x1; x2) , n = 1, 2 , which are algebraically solvable. Here l is the "discrete-time" independent variable taking integer values (l = 0, 1,…
In this article we have studied complex linear homogeneous difference equations where the coefficients are meromorphic functions, having finite iterated p-phi order. We have made some estimations on the growth of its nontrivial solutions.…
The problem of classification into symmetry integrable classes is solved for a family of second order nonlinear evolution equations labeled by arbitrary functions. Four nonequivalent symmetry integrable classes are thus obtained and the…
In this paper, we discuss the well-posedness of the Cauchy problem associated with the third-order evolution equation in time $$ u_{ttt} +A u + \eta A^{\frac13} u_{tt} +\eta A^{\frac23} u_t=f(u) $$ where $\eta>0$, $X$ is a separable Hilbert…
We consider the maximal regularity problem for non-autonomous evolution equations \begin{equation} \left\{ \begin{array}{rcl} u'(t) + A(t)\,u(t) &=& f(t), \ t \in (0, \tau] u(0)&=&u_0. \end{array} \right. \end{equation} Each operator $A(t)$…
We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[…
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow can not be computed exactly. Instead, we use a numerical…
We study a class of time-dependent (TD) non-Hermitian Hamiltonians $H(t)$ that can be transformed into a time-independent pseudo-Hermitian Hamiltonian $\mathcal{H}_{0}^{PH}$ using a suitable TD unitary transformation $F(t)$. The latter can…
We consider the bifurcation problem u'' + \lambda u = N(u) with two point boundary conditions where N(u) is a general nonlinear term which may also depend on the eigenvalue \lambda. A new derivation of a variational principle for the lowest…
One proves the $H$-theorem for mild solutions to a nondegenerate, nonlinear Fokker-Planck equation $$ u_t-\Delta\beta(u)+{\rm div}(D(x)b(u)u)=0, \ t\geq0, \ x\in\mathbb{R}^d,\qquad (1)$$ and under appropriate hypotheses on $\beta,$ $D$ and…