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 $L^1$ comparison estimate, uniqueness of the solution, and a minimality principle among a measure-free entropy supersolution class. For diffusion depending on both the solution and its gradient, we prove existence for continuous time-dependent obstacles with a boundary-controlled decomposition. Using two ordered penalization procedures, we obtain convergence of the approximate solutions and of the corresponding reaction terms. The limiting reaction consists of an absolutely continuous part with bounded density and a finite Radon measure concentrated on a prescribed compact spatial region. A one-sided time regularization yields strong convergence of the gradients and identifies the nonlinear flux. Together, the results yield existence, uniqueness and $L^1$ stability under their common assumptions.
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...
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 a double-logarithmic stability estimate for the recovery of a time-dependent potential in the parabolic equation $(c^{-1}\partial_t-\Delta_g+q)u=0$ on a conformally transversally anisotropic manifold. The partial input--output map associates an initial state and lateral Dirichlet data supported in a prescr...
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 prove non-uniqueness for the advection-diffusion equation with a drift in a logarithmically near-critical class below $L_t^2L_x^\infty$. The first construction is a diffusion-assisted inverse cascade of scalar energy that diverges as $t \to 0$. The second uses compressed inverse mixing, with diffusion acting perturb...
We study the large-time behavior of bounded entropy solutions to a class of fractional degenerate parabolic–hyperbolic equations involving Λα=(−Δ)α/2, 0<α<2, with initial data taken in generalized Besicovitch spaces associated with algebras possessing a mean value. The diffusion is governed by a nonlocal fractional ope...
Abid Khan, M. Z. Abidin, A. Algethami· Fractal and Fractional· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
