Related papers: On the Classifications of Scalar Evolution Equatio…
We consider a heat-type operator L structured on the left invariant 1-homogeneous vector fields which are generators of a Carnot group, multiplied by a uniformly positive matrix of bounded measurable coefficients depending only on time. We…
We report a class of symmetry-intergable third-order evolution equations in 1+1 dimensions under the condition that the equations admit a second-order recursion operator that contains an adjoint symmetry (integrating factor) of order six.…
Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…
We define various higher-order Markov properties for stochastic processes $(X(t))_{t\in \mathbb{T}}$, indexed by an interval $\mathbb{T} \subseteq \mathbb{R}$ and taking values in a real and separable Hilbert space $U$. We furthermore…
We show how the approach of Yosida approximation of the derivative serves to obtain new results for evolution systems. Using this method we obtain multivalued time dependent perturbation results. Additionally, translation invariant…
We consider differential-difference equations that determine the continuous symmetries of discrete equations on the triangular lattice. It is shown that a certain combination of continuous flows can be represented as a scalar evolution…
A generalized summation by parts algorithm is presented for solving of difference equations of the form $T^m(y)-a[u]y=b[u]$ where $T$ denotes the shift $u_j\to u_{j+1}$. Solvability of such type of equations with respect to coefficients of…
We study nonparametric estimation in dynamical systems described by ordinary differential equations (ODEs). Specifically, we focus on estimating the unknown function $f \colon \mathbb{R}^d \to \mathbb{R}^d$ that governs the system dynamics…
In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} \Delta^{2}u-M(\|\nabla u\|_{2}^{2})\Delta u+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N},…
We propose a new method for solution of the integrability problem for evolutionary differential-difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is…
The continuous dependence of solutions to certain (non-autonomous, partial, integro-differential-algebraic, evolutionary) equations on the coefficients is addressed. We give criteria that guarantee that convergence of the coefficients in…
Let us fix a prime $p$ and a homogeneous system of $m$ linear equations $a_{j,1}x_1+\dots+a_{j,k}x_k=0$ for $j=1,\dots,m$ with coefficients $a_{j,i}\in\mathbb{F}_p$. Suppose that $k\geq 3m$, that $a_{j,1}+\dots+a_{j,k}=0$ for $j=1,\dots,m$…
Logistic equations play a pivotal role in the study of any non linear evolution process exhibiting growth and saturation. The interest for the phenomenology, they rule, goes well beyond physical processes and cover many aspects of ecology,…
That the weak solutions of degenerate parabolic pdes modelled on the inhomogeneous $p-$Laplace equation $$ u_t - \mathrm{div} \left(|\nabla u|^{p-2} \nabla u \right) = f \in L^{q,r}, \quad p>2 $$ are $C^{0,\alpha}$, for some $\alpha \in…
We characterize the set of functions $u\_0\in L^2(R^n)$ such that the solution of the problem $u\_t=\mathcal{L}u$ in $R^n\times(0,\infty)$ starting from $u\_0$ satisfy upper and lower bounds of the form $c(1+t)^{-\gamma}\le \|u(t)\|\_2\le…
We discuss existence, uniqueness, and space-time H\"older regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where $A$ generates an…
In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R},…
There are considered differential substitutions of the form $v=P(x,u,u_{x})$ for which there exists a differential operator $H=\sum^{k}_{i=0} \alpha_{i} D^{i}_{x}$ such that the differential substitution maps the equation…
In this paper we investigate fractional differential equations with Hilfer fractional derivative of order $1<\gamma<2$ and type $\delta \in [0,1]$ in a Banach space. We introduce a family of general fractional cosine operator functions of…
The present paper is a continuation of our recent paper \cite{DaoReissig}. We will consider the following Cauchy problems for semi-linear structurally damped $\sigma$-evolution models: \begin{equation*} u_{tt}+ (-\Delta)^\sigma u+ \mu…