# A Necessary and Suffcient Condition for Uniqueness of the Trivial Solution in Semilinear Parabolic Equations

## TL;DR

This paper establishes that the uniqueness of the trivial solution in semilinear parabolic equations is directly linked to the Osgood-type condition on the nonlinearity, removing the need for the concavity assumption previously thought necessary.

## Contribution

It provides an elementary proof of non-uniqueness without the concavity condition, showing the equivalence between PDE and ODE solution uniqueness.

## Key findings

- Uniqueness of the trivial solution in PDEs is equivalent to the ODE case.
- The non-concavity assumption on the nonlinearity is unnecessary for non-uniqueness.
- The result simplifies understanding of solution uniqueness in semilinear parabolic equations.

## Abstract

In their 1968 paper Fujita and Watanabe considered the issue of uniqueness of the trivial solution of semilinear parabolic equations with respect to the class of bounded, non-negative solutions. In particular they showed that if the underlying ODE has non-unique solutions (as characterised via an Osgood-type condition) {\em and} the nonlinearity $f$ satisfies a concavity condition, then the parabolic PDE also inherits the non-uniqueness property. This concavity assumption has remained in place either implicitly or explicitly in all subsequent work in the literature relating to this and other, similar, non-uniqueness phenomena in parabolic equations. In this paper we provide an elementary proof of non-uniqueness for the PDE without any such concavity assumption on $f$. An important consequence of our result is that uniqueness of the trivial solution of the PDE is equivalent to uniqueness of the trivial solution of the corresponding ODE, which in turn is known to be equivalent to an Osgood-type integral condition on $f$.