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 parabolic $p$-Laplace equation with nonnegative time-independent forcing and Dirichlet data. By combining a time-rescaling argument with $L^\infty$ comparison estimates, we derive Lipschitz regularity in time after any fixed positive elapsed time, which reduces the problem to an elliptic-type decay estimate for the spatial measure of level sets that is essentially uniform in time. This reduction enables us to address the above obstacle.
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...
We establish the local boundedness of time derivatives of bounded viscosity solutions to the normalized parabolic $p$-Laplace equation for every $p\in(1,\infty)$. The proof combines a logarithmic Bernstein argument with quantitative stability to derive a recursive estimate between different regularizations. For $p=1$ a...
This paper establishes the exponential $L^2$-stability of the unique solutions to initial-boundary value problems for linear parabolic partial differential equations with general drift and zero-order coefficients in bounded domains. The key idea lies in constructing a suitable Dirichlet form with respect to a weighted...
We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional $n\times n$ Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of...
We establish an intrinsic Harnack inequality with an optimal tail term for weak solutions of parabolic fractional $p$-Laplace equations with $p>2$ and $s\in(0,1)$ whose kernels are symmetric, measurable, and comparable to $|x-y|^{-n-sp}$. It constitutes the nonlocal analogue of the intrinsic Harnack inequality of DiBen...
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 fr...
Cheng Yuan· 0 citations
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