Lee and Ni proved in the paper Global existence, large time behavior and life span of solutions of semi linear parabolic Cauchy problem that the heat semigroup decays as $t$ tends to infinity, that is $$\|S(t)u_0\|_{\infty} \sim t^{-n/2}$$ for initial data $u_0 \in C_b(\mathbb{R}^n)$ (space of all bounded continuous functions) or $u_0$ in spaces where $|x|^{\alpha}$ times $u_0(x)$ is bounded. In limited domains, exponential decay is known. My question is if there is some kind of initial data $u_0$ so that instead of decaying, the semigroup grows? For example $$\|S(t)u_0\|_{\infty} \sim t^{\alpha}, \,\,\, \alpha>0.$$ is there anything in that direction? The reason for the question is because I'm in a problem where I need to replace $S(t)u_0$ with $[S(t)u_0]^{-\alpha}$, with $\alpha>0.$ So I thought that if there is any possibility of growth in the semigroup, I can make this exchange.
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$\begingroup$ Please note that your claim about the decay of $\|S(t)u_0\|$ for $u_0 \in C_b$ is not correct: for instance, if $u_0$ is the constant function with value $1$, than $S(t)u_0 = u_0$ for all $t$, so the orbit does not decay at all. What the linked paper actually says (in Lemma 2.12(iii)) is that the solutions does not decay faster than $t^{-n/2}$; some solutions may decay slower, though. $\endgroup$– Jochen GlueckCommented Mar 18, 2022 at 11:08
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$\begingroup$ Actually, it seems that $\|S(t)u_0\|_\infty$ decays with rate $t^{-n/2}$ if $u_0 \gneq 0$ is in $L^1(\mathbb{R}^n) \cap C_b(\mathbb{R}^n)$. The lower estimate for the decay rate is in the paper you cited, and the upper estimate follows simply from the fact that the sup norm of the heat kernel satisfies this estimate. $\endgroup$– Jochen GlueckCommented Mar 18, 2022 at 11:20
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$\begingroup$ was a typo, I meant to refer to the $L^{\infty}$ norm, for bounded continuous positive data, in bounded or unbounded domains, $||S(t)u_0||_{\infty}$ decays . My question is if there is any data class where $||S(t)u_0||_{\infty}$ grows when $t\rightarrow \infty$. Do you know anything about it? $\endgroup$– IlovemathCommented Mar 18, 2022 at 13:21
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$\begingroup$ Thanks for your reply! Yes, I'm aware that you were referring to the $L^\infty$-norm, and in both my comments I was referring to the $L^\infty$-norm, too. As I said, it is not true that $\|S(t)u_0\|_\infty \sim t^{-n/2}$ for general $0 \lneq u_0 \in C_b(\mathbb{R}^n)$ (see the counterexample I mentioned in my first comment where the solution does not decay at all). $\endgroup$– Jochen GlueckCommented Mar 18, 2022 at 14:05
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$\begingroup$ I understand, thank you very much for the comment, it helps a lot! In this case, $||S(t)u_0||_{\infty}=1$. I was wondering if it could grow, for example $||S(t)u_0||_{\infty}=Ct^{\alpha}$, $\alpha>1$. Maybe that never happens, whenever I think of a function, the semigroup is unbounded in it. $\endgroup$– IlovemathCommented Mar 18, 2022 at 16:20
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