Timeline for Question on Parabolic PDEs: Improvement of the bound ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p}\le \hat C t^{1/q}$

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Jul 12, 2019 at 14:14	comment	added	Andrew		@kaithkolesidou Consider the Cauchy problem for the heat equation $u_t-\Delta u=0$ in $\mathbb R^n$. The bounded solution will be smooth for $t>0$. So uniforme Lipschitz $u_t$ wrt $t$ means boundedness of $u_{tt}$. From the equation $u_{tt}=\Delta^2u$. At the initial moment $t=0$ it turns into $u_{tt}=\Delta^2v_0$. So for $u_{tt}$ to be bounded in all half space $t>0$ it is necessary that $\Delta^2v_0\in L_\infty(\mathbb R^n)$.
Jul 12, 2019 at 11:56	comment	added	kaithkolesidou		@Andrew Will I need this kind of regularity for my initial data since I want $\partial_t u$ to be Lipschitz...? Is there any theorem I miss in this case?
Jul 11, 2019 at 16:36	comment	added	Andrew		@kaithkolesidou If you want in fact boundlessness of $u_t$ the initial function $v_0$ should be from $C^{1,1}$ (first order derivatives are Lipschitz) as illustrated by this post.
Jul 11, 2019 at 15:36	comment	added	kaithkolesidou		Maybe I should had written this in advance but $f$ depends on $v$ somehow, yet is still uniformly bounded in $L^{\infty}(M\times [0,T])$. This is why I didn't think of the fundamental solution. However, what is wrong in my approach?
Jul 11, 2019 at 15:28	history	answered	Bazin	CC BY-SA 4.0