# Lack of smoothing for bounded solutions of a semilinear parabolic equation

## TL;DR

This paper investigates a semilinear parabolic equation with globally bounded solutions that develop interior gradient singularities over time, converging to a stationary solution with similar singularity characteristics.

## Contribution

It provides new insights into the behavior of bounded solutions with interior gradient singularities in semilinear parabolic equations, highlighting the lack of smoothing effects.

## Key findings

- Solutions have interior gradient singularities for all positive times.
- Solutions converge to stationary solutions with similar singularities.
- The study clarifies the nature of singularities in bounded solutions.

## Abstract

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as $t\to\infty$.