About the existence and uniqueness theorem for hyperbolic equation

Picard’s Existence and Uniqueness Theorem

One of the most important theorems in Ordinary Di↵erential Equations is Picard’s Existence and Uniqueness Theorem for first-order ordinary di↵erential equations. Why is Picard’s Theorem so important? One reason is it can be generalized to establish existence and uniqueness results for higher-order ordinary di↵erential equations and for systems of di↵erential equations. Another is that it is a g...

Existence and Uniqueness of the Entropy Solution to a Nonlinear Hyperbolic Equation

This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: u t + div(vf (u)) = 0 in IR N 0; T ], with initial data u(:; 0) = u 0 (:) in IR N. where u 0 2 L 1 (IR N) is a given function, v is a divergence-free bounded function of class C 1 from IR N 0; T ] to IR N , and f is a function of class C 1 from IR to...

Existence and Uniqueness for a Variational Hyperbolic System without Resonance

Existence and Uniqueness Theorem for Fractional Differential Equation with Integral Boundary Condition

We have investigated the existence and uniqueness solutions of the nonlinear fractional differential equation of an arbitrary order with integral boundary condition. The result is an application of the Schauder fixed point theorem and the Banach contraction principle.

Optimal Existence and Uniqueness Theory for the Fractional Heat Equation

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation ∂tu + (−∆) s u = 0, 0 < s < 1, posed in the whole space R with data in a class of locally bounded Radon measures that are allowed to grow at infinity with an optimal growth rate. We consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative d...