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SIMIODE EXPO 2021 - Minicourse M-R2 - Delay Differential Equations In Epidemiology
Full text linkMany problems in epidemiology give rise to delay differential equations (DDEs). These are differential equations in which the current rate of change of the system depends not only on the current state but also on the history of the system; i.e. the system has memory. In this minicourse, we will discuss some key tools necessary to understand the applications involving DDEs. We will go through an example that illustrates the need and implementation of DDE in certain infectious diseases
Nonlinear Second Order Parabolic and Elliptic Equations With Nonlinear Boundary Conditions
Full text linkSteklov Spectrum And Elliptic Problems With Nonlinear Boundary Conditions
No full textProblems with nonlinear boundary conditions arise naturally in many applications. For instance, in population dynamics where an impact of habitat-edges (boundary) on the dispersal pattern of species as they reach the boundary takes place in spatial ecology CC06. They occur when the biochemical reactions take place at or near the boundary, for example, in the limb bud development of a chick in which a chemical reaction produces outgrowth due to cell growth and division, and interactions between morphogens produced in several zones of the limb bud DO99. They also appear in noninvasive testing methods to locate defects in a medium by using boundary data measurements (see, e.g., CCMM16). In cryosurgery (a minimally invasive treatment used to treat some types of cancers and some conditions that may become cancer), a highly exothermic reaction takes place in a thin layer around the boundary in order to destroy abnormal tissue LOS98. These examples are not exhaustive
Generalized Eigenproblem And Nonlinear Elliptic Equations With Nonlinear Boundary Conditions
Full text linkWe are concerned with the solvability of nonlinear second-order elliptic partial differential equations with nonlinear boundary conditions. We study the generalized Steklov-Robin eigenproblem (with possibly singular weights) in which the spectral parameter is both in the differential equation and on the boundary. We prove the existence of solutions for nonlinear problems when both nonlinearities in the differential equation and on the boundary interact, in some sense, with the generalized spectrum. The proofs are based on variational methods and a priori estimates
Equivalence Between Uniform Lp∗ A Priori Bounds and Uniform L∞ A Priori Bounds for Subcritical p-Laplacian Equations
No full textWe establish sufficient conditions for a uniform Lp⋆(Ω) bound to imply a uniform L∞(Ω) bound for positive weak solutions of subcritical p-Laplacian equations. We also provide an equivalent result for sequences of boundary-value problems. As consequences, we prove that any set of solutions with finite energy is L∞(Ω) a priori bounded, and also obtain an alternative proof of the existence of a priori bounds for subcritical power like nonlinearities
A Priori Bounds And Existence Of Positive Solutions For Semilinear Elliptic Systems
Full text linkWe provide a-priori L∞ bounds for classical positive solutions of semilinear elliptic systems in bounded convex domains when the nonlinearities are below the power functions v^p and u^q for any (p,q) lying on the critical Sobolev hyperbola. Our proof combines moving planes method and Rellich–Pohozaev type identities for systems. Our analysis widens the known ranges of nonlinearities for which classical positive solutions of semilinear elliptic systems are a priori bounded. Using these a priori bounds, and local and global bifurcation techniques, we prove the existence of positive solutions for a corresponding parametrized semilinear elliptic system
Strong Bounded Solutions For Nonlinear Parabolic Systems
Full text linkIn this article we study the existence of strong bounded solutions for nonlinear parabolic systems on a domain which is bounded in space and unbounded in time (namely the entire real line). We use nonlinear iteration arguments combined with some a priori estimates to derive the existence results. We also provide conditions under which we have a positive solution. Some examples are given to illustrate the results
Resonance Problems For Nonlinear Elliptic Equations With Nonlinear Boundary Conditions
Full text linkWe study the solvability of nonlinear second order elliptic partial differential equations with nonlinear boundary conditions where we impose asymptotic conditions on both nonlinearities in the differential equation and on the boundary in such a way that resonance occurs at a generalized eigenvalue; which is an eigenvalue of the linear problem in which the spectral parameter is both in the differential equation and on the boundary. The proofs are based on some variational techniques and topological degree arguments
Nonresonance On The Boundary And Strong Solutions Of Elliptic Equations With Nonlinear Boundary Conditions
Full text linkWe deal with the solvability of linear second order elliptic partial differential equations with nonlinear boundary conditions by imposing asymptotic nonresonance conditions of nonuniform type with respect to the Steklov spectrum on the boundary nonlinearity. Unlike some recent approaches in the literature for problems with nonlinear boundary conditions, we cast the problem in terms of nonlinear compact perturbations of the identity on appropriate trace spaces in order to prove the existence of strong solutions. The proofs are based on a priori estimates for possible solutions to a homotopy on suitable trace spaces and topological degree arguments
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