A two dimensional stochastic differential equation is suggested as a stochastic model for the Kermack-McKendrick epidemics. Its strong (weak) existence and uniqueness and absorption properties are investigated. The examples presented in Section 5 are meant to illustrate possible different asymptotics of a solution to the equation.
Over a large range of the pressure, one cannot ignore the fact that the viscosity grows significantly (even exponentially) with increasing pressure. This paper concerns long-time and large-data existence results for a generalization of the Navier-Stokes fluid whose viscosity depends on the shear rate and the pressure. The novelty of this result stems from the fact that we allow the viscosity to be an unbounded function of pressure as it becomes infinite. In order to include a large class of viscosities and in order to explain the main idea in as simple a manner as possible, we restrict ourselves to a discussion of the spatially periodic problem.
This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field. The analysis is performed on a time-dependent domain which characterizes the nondegenerated elastic material regions. We choose a notion of weak solutions which consists of weak formulations of the Cahn-Hilliard system and the momentum balance equation, a variational inequality for the damage evolution and an energy inequality. For the introduced degenerating system, we prove global-in-time existence of weak solutions. The main results are sketched from our recent paper [WIAS preprint no. 1759 (2012)].
We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of $p$-coercivity and $(p-1)$-growth, for a given parameter $p > 1$. The existence of Dirichlet weak solutions was obtained in [2], in the cases $p \ge 12/5$ if $d = 3$ or $p \ge 2$ if $d = 2$, $d$ being the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all $p \ge 2$. In addition, we obtain regularity properties of weak solutions whenever $p \ge 20/9$ (if $d = 3$) or $p \ge 2$ (if $d = 2$). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained.
We consider the following quasilinear Neumann boundary-value problem of the type − ∑ N i=1 ∂ ⁄ ∂xi ai ( x, ∂u ⁄ ∂xi ) + b(x)|u| p0(x)−2 u = f(x, u) + g(x, u) in Ω, ∂u ⁄ ∂γ = 0 on ∂Ω. We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.
We study the existence and the uniqueness of the weak solution of an inverse problem for a semilinear higher order ultraparabolic equation with Lipschitz nonlinearity. The main aim is to determine the weak solution of the equation and some functions that depend on the time variable, appearing on the right-hand side of the equation. The overdetermination conditions introduced are of integral type. In order to prove the solvability of this problem in Sobolev spaces we use the Galerkin method and the method of successive approximations.
This paper proposes a stochastic diffusion model for the spread of a susceptible-infective- removed Kermack–McKendric epidemic (M1) in a population which size is a martingale Nt that solves the Engelbert–Schmidt stochastic differential equation (2). The model is given by the stochastic differential equation (M2) or equivalently by the ordinary differential equation (M3) whose coeffients depend on the size Nt. Theorems on a unique strong and weak existence of the solution to (M2) are proved and computer simulations performed.
We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)<q(x)< \frac{N\cdot m(x)}{(N-m(x))}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda $ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}} _2$-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.
We consider the two-dimesional spatially periodic problem for an evolutionary system describing unsteady motions of the fluid with shear-dependent viscosity under general assumptions on the form of nonlinear stress tensors that includes those with pstructure. The global-in-time existence of a weak solution is established. Some models where the nonlinear operator corresponds to the case p = 1 are covered by this analysis.
We are concerned with the problem of differentiability of the derivatives of order m + 1 of solutions to the “nonlinear basic systems” of the type {\left( { - 1} \right)^m}\sum\limits_{\left| \alpha \right| = m} {{D^\alpha }{A^\alpha }\left( {{D^{\left( m \right)}}u} \right)} + \frac{{\partial u}}{{\partial t}} = 0\;in\;Q. We are able to show that {D^\alpha }u \in {L^2}\left( { - a,0,{H^\partial }\left( {B\left( \sigma \right),{\mathbb{R}^N}} \right)} \right),\;\left| \alpha \right| = m + 1, for \partial\in \left ( 0,1/2 \right )and this result suggests that more regularity is not expectable., Roberto Amato., and Obsahuje seznam literatury