Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation

TL;DR

This paper investigates the global existence, uniqueness, and growth behavior of positive solutions to a sublinear pseudoparabolic equation with unbounded, time-dependent potential, providing key bounds and asymptotic rates.

Contribution

It establishes global existence, uniqueness, and precise grow-up rates for solutions with various initial conditions, including unbounded and power radial growth.

Findings

Solutions exhibit a lower grow-up and radial growth bound.

A comparison principle is proved for the equation.

The asymptotic grow-up rate and critical growth exponent are derived.

Abstract

We study positive solutions of the pseudoparabolic equation with a sublinear source in $\mathbb{R}^n$. In this work, the source coefficient could be unbounded and time-dependent. Global existence of solutions to the Cauchy problem is established within weighted continuous spaces by approximation and monotonicity arguments. Every solution with non-zero initial value is shown to exhibit a certain lower grow-up and radial growth bound, depending only upon the sublinearity and the unbounded, time-dependent potential. Using the lower grow-up/growth bound, we can prove the key comparison principle. Then we settle the uniqueness of solutions for the problem with non-zero initial condition by employing the comparison principle. For the problem with the zero initial condition, we can classify the non-trivial solutions in terms of the maximal solutions. When the initial condition has a power radial growth, we can derive the precise asymptotic grow-up rate of solutions and obtain the critical growth exponent.