Doubly Exponential Growth and Decay for a Semilinear Heat Equation With Logarithmic Nonlinearity

In this note, we consider the initial boundary value problem for a parabolic equation with logarithmic nonlinearity, which has been studied by Chen et al. and Han. We firstly estimate the upper bound function with doubly exponential property for the weak solutions and establish the doubly exponential decay and at most doubly exponential growth criteria for the weak solutions by the logarithmic Sobolev inequality, the derivative formula for the product, the Newton-Leibniz formula, and the logarithmic properties. We secondly establish a newly infinity blow-up criterion such that the lower bound function of weak solutions is at least doubly exponential growth by the nonconcavity method and the nonincreasing property for the energy functional. We finally establish a sufficient condition such that the upper bound function with doubly exponential growth is greater than or equal to the lower bound function with doubly exponential growth by the quotient comparison principle. In addition, we also find the threshold of the doubly exponential growth and the doubly exponential decay and obtain & Vert;u & Vert;22 -> 0+$$ {\left\Vert u\right\Vert}_2<^>2\to {0}<^>{+} $$ for any t >= 0$$ t\ge 0 $$ as & Vert;u0 & Vert;22 -> 0+.$$ {\left\Vert {u}_0\right\Vert}_2<^>2\to {0}<^>{+}. $$ These research results improve previous results from both decay and growth.