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For a general linear parabolic equation, is a strong maximum principle possible when the solutions are merely weak solutions (i.e. they lie in a Bochner space)? Is there some proof possible that does not use classical theory/solutions?

Note that I also posted this on MathOverflow https://mathoverflow.net/questions/179654/strong-maximum-principle-for-weak-solutions.

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As far as I know, in general case the answer is negative. Even for the problem $$ u'_t = \mbox{div}(A(x,t) \nabla u) + f(x,t) $$ (with suitable initial-boundary conditions) the Strong maximum principle is unknown if $A(x,t)$ is only continuous (not smooth). (In this case you cannot use the classical technique of comparison with some "good" function, since you cannot expand the divergence for it.)

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  • $\begingroup$ If $A$ is smooth (eg. $A\equiv 1$) how does one do the proof using just the setting of weak solutions? $\endgroup$ – LapLace Sep 3 '14 at 16:14
  • $\begingroup$ @LapLace Unfortunately, I don't know it. $\endgroup$ – Voliar Sep 3 '14 at 16:21

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