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A new proof of a fundamental supnorm estimate for one-dimensional advection-diffusion equations. (English) Zbl 1372.35149
Summary: We give a short derivation of supnorm estimates for solutions of one-dimensional advection-diffusion equations \[ u_t+f(u)_=u_{xx},\qquad x\in\mathbb R,\;t>0 \] and some of their generalizations, assuming initial data \(u(\cdot,0)\in L^p (\mathbb R)\cap L^\infty (\mathbb R)\) for some \(1\leq p<\infty\). A few related results and open questions are also given.
MSC:
35K57 Reaction-diffusion equations
35K59 Quasilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
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