About symmetric rank-1 random matrices

fa.practical evaluation – Does $int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le C$ indicate $Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C$ within the warmth equation? Answer

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fa.practical evaluation – Does $int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le C$ indicate $Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C$ within the warmth equation?

For the parabolic equation
$$u_t + f(u)_x – u_{xx} = 0$$
one has
$$Vert u(t,cdot) Vert_{L^2(mathbb R)} + 2int_0^t Vert u_x(s,cdot) Vert_{L^2} ds le Vert u(0,cdot) Vert_{L^2(mathbb R)}.$$
If $t le T$ (for a hard and fast $T>0$), does the appraise above indicate too $$Vert u_x (t,cdot) Vert_{L^2(mathbb R)} le C$$
(with out time integration)?

In different phrases, one can too question: if $int_0^T f(s) ds le C$ is it undoubted that $f(t) le C$ for all $t in [0,T]$?

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