The presented work investigates the Cauchy problems for parabolic equations in both non-divergence and divergence forms with rough diffusion coefficients, which commonly arise in composite media, financial pricing, and viscous fluids. Under critical regularity settings, we establish the unique solvability in optimal fractional Sobolev spaces. The first key technical step lies in constructing an effective approximation scheme with truncated and mollified diffusion coefficients $a^\epsilon(t,x)$, for which we rigorously prove the uniform preservation of high-frequency smallness. By incorporating this approximation scheme with paraproduct decomposition, Fefferman-Stein maximal inequalities, Coifman-Meyer bilinear estimates, and refined Sobolev embeddings, we close the uniform a priori estimates and then pass to the limit. Furthermore, we prove that the threshold $s<\frac{1}{2}$ is sharp by constructing explicit counterexamples. These theoretical results provide a rigorous mathematical framework in the study of Cauchy problems for parabolic equations with rough coefficients.
For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their H\"{o}lder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcom...
This paper studies doubly nonlinear parabolic obstacle problems. We introduce a renormalized entropy formulation in which the reaction measure is encoded by evaluating the entropy multiplier at the obstacle. For diffusion depending only on the gradient and continuous time-independent obstacles, we prove a global $L^1$...
In this paper, we consider a nonlinear fractional superdiffusion equation involving a Caputo derivative of order α∈(1,2). Solutions to this class of problems typically exhibit weak singularities near the initial time. To handle this singularity, we introduce an auxiliary variable p:=Dtα/2(u−tu1) and reformulate the ori...
We study the initial–boundary value problem for a fractional nonlinear Klein–Gordon equation posed on the hyperoctant Dn,n⩾2 with nonhomogeneous boundary data on each coordinate hyperplane. The equation involves a spatial fractional Laplacian of Caputo type with order β∈(3/2,2), which induces strong nonlocal dissipatio...
It is known that a major difficulty in proving H\"{o}lder regularity for solutions to quasilinear singular parabolic equations of $p$-Laplace type via the method of intrinsic scaling is to establish the decay estimate of the space-time measure of level sets. In this paper, we consider an inhomogeneous singular paraboli...
We study three classical problems in the theory of linear second-order uniformly elliptic equations: (i) gradient estimates for the Dirichlet problem, (ii) estimates and asymptotics for the first eigenvalue of an elliptic operator, and (iii) the Landis conjecture on exponential decay. Our main results give versions of...
B. Sirakov, Philippe Souplet· 0 citations
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