Weak Solutions Of Parabolic Equations Research Articles

Time Regularity for Local Weak Solutions of the Heat Equation on Local Dirichlet Spaces

We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a very weak L2 Gaussian type upper bound for the heat semigroup, we prove that the time derivatives of a local weak solution of the heat equation are themselves local weak solutions. This applies, for instance, to local weak solutions of parabolic equations with uniformly elliptic symmetric divergence form second order operators with measurable coefficients. We describe some applications to the structure of ancient local weak solutions of such equations which generalize recent results of Colding and Minicozzi (Duke Math. J., 170(18), 4171–4182 2021) and Zhang (Proc. Amer. Math. Soc., 148(4), 1665–1670 2020).

The regularity of the positive part of functions in $L^2(I; H^1(\Omega)) \cap H^1(I; H^1(\Omega)^*)$ with applications to parabolic equations

Let $u\in L^2(I; H^1(\Omega))$ with $\partial_t u\in L^2(I; H^1(\Omega)^*)$ be given. Then we show by means of a counter-example that the positive part $u^+$ of $u$ has less regularity, in particular it holds $\partial_t u^+ \notin L^1(I; H^1(\Omega)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

Optimal Gradient Estimates for Parabolic Equations in Variable Exponent Spaces

We establish an optimal |$L^{p(\cdot )}$|-regularity theory for the gradient of weak solutions for parabolic equations in divergence form with bounded measurable coefficients in rough domains beyond the Lipschitz category. With a function |$p(\cdot )=p(x,t)$| of spatial and time variables satisfying log-Hölder continuity, we prove that the spatial gradient of the weak solution is as integrable as the nonhomogeneous term in the variable exponent space |$L^{p(\cdot )}$| under the assumption that the coefficients are merely measurable in one of spatial variables while have a small bounded mean oscillation in the other spatial variables and time variable. On the other hand, the domain is assumed to be sufficiently flat in the Reifenberg sense.

Study of weak solutions for parabolic equations with nonstandard growth conditions

The authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for u t = div ( ( u σ + d 0 ) | ∇ u | p ( x , t ) − 2 ∇ u ) + f ( x , t ) . Localization property of weak solutions is also discussed.

Optimal W1,p regularity theory for parabolic equations in divergence form

This work treats Lp regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains. We essentially obtain an optimal condition on the coefficients under which the global W1,p regularity theory holds.

Hölder Continuity of Weak Solutions for Parabolic Equations with Nonstandard Growth Conditions

In this paper, we investigate the interior regularity including the local boundedness and the interior Holder continuity of weak solutions for parabolic equations of the p(x, t)–Laplacian type. We improve the Moser iteration technique and generalize the known results for the elliptic problem to the corresponding parabolic problem.

Hölder continuity of local weak solutions for parabolic equations exhibiting two degeneracies

We consider equations of the form $${\partial}_{t} v - \mbox{div} ( \alpha (v) \nabla v) = 0 \ , $$ where $v \in [0,1]$ and $\alpha (v)$ degenerates for $v=0$ and $v=1$. We show that local weak solutions are locally Hölder continuous provided $\alpha$ behaves like a power near the two degeneracies. We adopt the technique of intrinsic rescaling developed by DiBenedetto.

Weak solutions of parabolic equations in non-cylindrical domains

In their classical work, Ladyzhenskaya and Ural ′ ’ tseva gave a definition of weak solution for parabolic equations in cylindrical domains. Their definition was broad enough to guarantee the solvability of all such problems but narrow enough to guarantee the uniqueness of these solutions. We give here some alternative definitions which are appropriate to non-cylindrical domains, and we prove the unique solvability of such problems.

Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients

Pointwise differentiability of weak solutions of parabolic equations with measurable coefficients

Wiener estimates at boundary points for parabolic equations

We prove an estimate on the modulus of continuity at a boundary point (Wiener estimate) for the weak solutions of parabolic equations.

On behaviour of non-negative weak solutions of parabolic equations at the boundary

On behaviour of non-negative weak solutions of parabolic equations at the boundary

A Property of Weak Solutions for Some Parabolic Equations of Higher Order

Recently Aronson [1] proved the uniqueness property of weak solutions of the initial boundary value problem for second order parabolic equations with discontinuous coefficients. An analogous result to Aronson’s was proved by Kuroda M in the case of some parabolic equations of higher order, where the method due to Aronson [2] plays an essential role. In this paper, under the same idea we shall be concerned with the asymptotic behavior of weak solutions for parabolic equations of higher order of the divergence form, when the data are prescribed on a portion of a time-like surface.

Note on the Uniqueness Proprety of Weak Solutions of Parabolic Equations

Aronson proved, in his paper [1], the existence and the uniqueness property of weak solutions of the initial boundary value problem for parabolic equations of second order with measurable coefficients. On the uniqueness of solutions of the Cauchy problem for such equations he also gave some interesting results in [2].