Classical and weak solutions of the partial differential equations associated with a class of two-point boundary value problems

Ning Ma,Zhen Wu

doi:10.1186/s13661-018-1040-9

Abstract

This paper is concerned with a kind of first-order quasilinear parabolic partial differential equations associated with a class of ordinary differential equations with two-point boundary value problems. We prove that the function given by the solution of an ordinary differential equation is the unique solution of a first-order quasilinear parabolic partial differential equation in both classical and weak senses.

Highlights

IntroductionWe study the problem of solving the following first-order quasilinear parabolic partial differential equation (PDE):
In this paper, we study the problem of solving the following first-order quasilinear parabolic partial differential equation (PDE): ⎧⎨∂tu(t, x) + xu(t, x)b(t, x, u(t, x)) + f (t, x, u(t, x)) = 0, ⎩u(T, x) = h(x), (t, x) ∈ [0, T] × Rn, (1) where xu = ( ∂u ∂ xi)1≤i≤n is an n-dimensional row vector
6 Conclusion In this paper, to our best knowledge, we are the first to study this kind of PDE systems associated with the two-point boundary value problems

Summary

Introduction

We study the problem of solving the following first-order quasilinear parabolic partial differential equation (PDE):. PDE (1) should be related to a family of coupled ordinary differential equations (ODEs) associated with a kind of two-point boundary problems parameterized by (t, x) ∈ [0, T] × Rn as follows:. This two-point boundary value problem can be embedded into an optimal control problem when applying the maximum principle; the existence and uniqueness results were. By some analysis techniques of those related references, in this paper, we study PDE (1) in both classical and weak senses, including a Sobolev weak solution and viscosity solution of the two-point boundary value problem. Sup |Ys|2 ≤ C|X T |2 + C |Xs|2 + α(s) 2 ds

C U1 – λ

Conclusion