The Maximum Principle for Systems of Parabolic Equations Subject to an Avoidance Set
نویسندگان
چکیده
Hamilton’s maximum principle for systems states that given a reaction-diffusion equation (semi-linear heat-type equation) for sections of a vector bundle over a manifold, if the solution is initially in a subset invariant under parallel translation and convex in the fibers and if the ODE associated to the PDE preserves the subset, then the solution remains in the subset for positive time. We generalize this result to the case where the subsets are time-dependent and where there is an avoidance set from which the solution is disjoint. In applications the existence of an avoidance set can sometimes be used to prove the preservation of a subset of the vector bundle by the PDE.
منابع مشابه
A new positive definite semi-discrete mixed finite element solution for parabolic equations
In this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. In the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations. Also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. Error estimates were also obtaine...
متن کاملA note on critical point and blow-up rates for singular and degenerate parabolic equations
In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...
متن کاملThe use of inverse quadratic radial basis functions for the solution of an inverse heat problem
In this paper, a numerical procedure for an inverse problem of simultaneously determining an unknown coefficient in a semilinear parabolic equation subject to the specification of the solution at an internal point along with the usual initial boundary conditions is considered. The method consists of expanding the required approximate solution as the elements of the inverse quadrati...
متن کاملPontryagin Maximum Principle for Semilinear and Quasilinear Parabolic Equations with Pointwise State Constraints
This paper studies the rst order necessary conditions for the optimal controls of semilinear and quasilinear parabolic partial di erential equations with pointwise state constraints. Pontryagin type maximum principle is obtained.
متن کاملGradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems
We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some “asymmetric sy...
متن کامل