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Questions about partial differential equations of parabolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
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Question on Parabolic PDEs: Improvement of the bound ${\vert \vert v(\cdot,t)-v_0(\cdot) \ve...
Maybe I'm missing something, but it doesn't seem that your desired inequality can be true (even with the right-hand side of $(*)$ replaced by $C t^{\epsilon}$). It would imply uniform convergence of $ …