Solutions Of Quasilinear Parabolic Equations Research Articles

Unified approach to $C^{1,\alpha}$ regularity for quasilinear parabolic equations

In this paper, we are interested in obtaining a unified approach to $C^{1,\alpha}$ estimates for weak solutions of quasilinear parabolic equations, the prototype example being $$ u\_t - \operatorname{div} \big(|\nabla u|^{p-2} \nabla u\big) = 0. $$ without having to consider the singular and degenerate cases separately. This is achieved via a new scaling and a suitable adaptation of the covering argument developed by E. DiBenedetto and A. Friedman.

On Periodic Solutions of Quasilinear Parabolic Equations with Boundary Conditions of Bitsadze–Samarskii Type

On Periodic Solutions of Quasilinear Parabolic Equations with Boundary Conditions of Bitsadze–Samarskii Type

On the continuity of solutions of quasilinear parabolic equations with generalized Orlicz growth under non-logarithmic conditions

We prove the continuity of bounded solutions for a wide class of parabolic equations with (p, q)-growth $$\begin{aligned} u_{t}-\mathrm{div}\left( g(x,t,|\nabla u|) \,\frac{\nabla u}{|\nabla u|}\right) =0, \end{aligned}$$ under the generalized non-logarithmic Zhikov’s condition $$\begin{aligned}g(x,t,\mathrm{v}/r)&\leqslant c(K)\,g(y,\tau ,\mathrm{v}/r), \quad (x,t), (y,\tau )\in Q_{r,r}(x_{0},t_{0}), \quad 0<\mathrm{v}\leqslant K\lambda (r),\\ &\quad\lim \limits _{r\rightarrow 0}\lambda (r)=0, \quad \lim \limits _{r\rightarrow 0} \frac{\lambda (r)}{r}=+\infty , \quad \int _{0} \lambda (r)\,\frac{dr}{r}=+\infty . \end{aligned}$$ In particular, our results cover new cases of double phase and degenerate double-phase parabolic equations.

Radon measure-valued solutions of quasilinear parabolic equations

We discuss some recent results concerning Radon measure-valued solutions of the Cauchy–Dirichlet problem for $\\partial_t u = \\Delta\\phi(u)$. The function $\\phi$ is continuous, nondecreasing, with a growth at most powerlike. Well-posedness and regularity results are described, which depend on whether the initial data charge sets of suitable capacity (determined both by the Laplacian and by the growth order of $\\phi$), and on suitable compatibility conditions at $\\pm\\infty$.

Expanding solutions of quasilinear parabolic equations

<p style='text-indent:20px;'>By using the theory of maximal <inline-formula><tex-math id="M1">\begin{document}$ L^{q} $\end{document}</tex-math></inline-formula>-regularity and methods of singular analysis, we show a Taylor's type expansion–with respect to the geodesic distance around an arbitrary point–for solutions of quasilinear parabolic equations on closed manifolds. The powers of the expansion are determined explicitly by the local geometry, whose reflection to the solutions is established through the local space asymptotics.

Behavior of blow-up solutions for quasilinear parabolic equations

We study the quasilinear parabolic equation (|u|q − 1u)t − Δpu = 0 in a multidimensional domain (0; T) × Ω under the condition u(t; x) = f(t; x) on (0; T) × 𝜕Ω, where the boundary function f blows-up at a finite time T, i.e., f(t; x) → ∞ as t → T. For p ⩾ q > 0 and the boundary function f with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where p > q to p = q is investigated. A general approach within the method of energy estimates to such problems is described.

Behavior of blow-up solutions for quasilinear parabolic equations

We study the quasilinear parabolic equation $(|u|^{q-1}u)_t-\Delta_p\,u=0$ in a multidimensional domain $(0,T)\times\Omega$ under the condition $u(t,x)=f(t,x)$ on $(0,T)\times\partial\Omega$, where the boundary function $f$ blows-up at a finite time $T$, i.e., $f(t,x)\rightarrow\infty$ as $t\rightarrow T$. For $p\geqslant q&gt;0$ and the boundary function $f$ with power-like behavior, the upper bounds of weak solutions of the problem are obtained. The behavior of solutions at the transition from the case where $p&gt;q$ to $p=q$ is investigated. A general approach within the method of energy estimates to such problems is described.

Uniform boundedness for weak solutions of quasilinear parabolic equations

In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form \[ u t − div ⁡ A ( x , t , ∇ u ) = 0 , u_t - \operatorname {div} \mathcal {A}(x,t,\nabla u) = 0, \] where the nonlinearity A ( x , t , ∇ u ) \mathcal {A}(x,t,\nabla u) is modelled after the well-studied p p -Laplace operator. The question of boundedness has received a lot of attention over the past several decades with the existing literature showing that weak solutions in either 2 N N + 2 &gt; p &gt; 2 \frac {2N}{N+2}&gt;p&gt;2 , p = 2 p=2 , or 2 &gt; p 2&gt;p , are bounded. The proof is essentially split into three cases mainly because the estimates that have been obtained in the past always included an exponent of the form 1 p − 2 \frac {1}{p-2} or 1 2 − p \frac {1}{2-p} , which blows up as p → 2 p \rightarrow 2 . In this note, we prove the boundedness of weak solutions in the full range 2 N N + 2 &gt; p &gt; ∞ \frac {2N}{N+2} &gt; p &gt; \infty without having to consider the singular and degenerate cases separately. Subsequently, in a slightly smaller regime of 2 N N + 1 &gt; p &gt; ∞ \frac {2N}{N+1} &gt; p &gt; \infty , we also prove an improved boundedness estimate.

Propagation of Singularities for Large Solutions of Quasilinear Parabolic Equations

Propagation of Singularities for Large Solutions of Quasilinear Parabolic Equations

Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts

AbstractThis paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

Interior and boundary higher integrability of very weak solutions for quasilinear parabolic equations with variable exponents

We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the form ut−divA(x,t,∇u)=0onΩ×(−T,T),u=0on∂Ω×(−T,T),where the non-linear structure A(x,t,∇u) is modeled after the variable exponent p(x,t)-Laplace operator given by |∇u|p(x,t)−2∇u. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary by constructing a suitable test function which is Lipschitz continuous and preserves the boundary values. In the interior case, such a result was proved in Verena Bögelein and Qifan Li (2014) provided p(x,t)≥p−≥2 holds and was then extended to the singular case 2nn+2<p−≤p(x,t)≤p+≤2 in Qifan Li (2017). This restriction was necessary because the intrinsic scalings for quasilinear parabolic problems are different in the case p+≤2 andp−≥2.In this paper, we develop a new approach, using which, we are able to extend the results of Verena Bögelein and Qifan Li (2014), Qifan Li (2017) to the full range 2nn+2<p−≤p(x,t)≤p+<∞ and also obtain analogous results up to the boundary. The main novelty of this paper is that we make use of a unified intrinsic scaling using which our methods are able to handle both the singular case and degenerate case simultaneously. Our techniques improve and simplify many aspects of the method of parabolic Lipschitz truncation (even in the constant exponent case) studied extensively in existing literature. To simplify the exposition, we will only prove the higher integrability result near the boundary, provided the domain Ω satisfies a uniform measure density condition and are non perturbative in nature, hence we make no regularity assumptions for the coefficients of the nonlinear operator. Our techniques are also applicable to higher order equations as well as systems.

Boundary higher integrability for very weak solutions of quasilinear parabolic equations

We prove boundary higher integrability for the (spatial) gradient of very weak solutions of quasilinear parabolic equations of the formut−divA(x,t,∇u)=0onΩ×R, where the non-linear structure divA(x,t,∇u) is modelled after the p-Laplace operator. To this end, we prove that the gradients satisfy a reverse Hölder inequality near the boundary. In order to do this, we construct a suitable test function which is Lipschitz continuous and preserves the boundary values. These results are new even for linear parabolic equations on domains with smooth boundary and make no assumptions on the smoothness ofA(x,t,∇u). These results are also applicable for systems as well as higher order parabolic equations.

Consistent two-sided estimates for the solutions of quasilinear parabolic equations and their approximations

For a linearized finite-difference scheme approximating the Dirichlet problem for a multidimensional quasilinear parabolic equation with unbounded nonlinearity, we establish pointwise two-sided solution estimates consistent with similar estimates for the differential problem. These estimates are used to prove the convergence of finite-difference schemes in the grid L 2 norm.

Global and blow-up solutions for quasilinear parabolic equations with a gradient term and nonlinear boundary flux

Abstract This work is concerned with positive classical solutions for a quasilinear parabolic equation with a gradient term and nonlinear boundary flux. We find sufficient conditions for the existence of global and blow-up solutions. Moreover, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’ and an upper estimate of the global solution are given. Finally, some application examples are presented. MSC:35R45, 35K65, 34A12.

Global existence and nonexistence of solutions for quasilinear parabolic equation

Abstract This work is concerned with the global existence and nonexistence of solutions for a quasilinear parabolic equation with null Dirichlet boundary condition. Based on the Galerkin approximation technique and the theory of a family of potential wells, we obtain the invariant sets and vacuum isolating of global solutions including critical case, and we also give global nonexistence. MSC:35A01, 35B06, 35B08.

On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations

For a broad class of initial-boundary value problems for quasilinear parabolic equations with nonlinear source and their approximations, we show that if the initial energy is negative, then the solution always blows up in finite time. This is especially important for finding sufficiently simple and easy-to-verify conditions guaranteeing the presence of physical effects such as heat localization in peaking modes or thermal explosions and for deriving two-sided estimates of the solution lifespan. We construct the corresponding new classes of difference schemes for which grid analogs of integral conservation laws hold. We show that, to obtain efficient two-sided estimates for the blow-up time of the solution of the differential problem, in practice, one should use difference schemes with explicit as well as implicit approximation to the source.

Initial evolution of supports of solutions of quasilinear parabolic equations with degenerate absorption potential

The propagation of supports of solutions of second-order quasilinear parabolic equations is studied; the equations are of the type of nonstationary diffusion, having semilinear absorption with an absorption potential which degenerates on the initial plane. We find sufficient conditions, which are sharp in a certain sense, on the relationship between the boundary regime and the type of degeneration of the potential to ensure the strong localization of solutions. We also establish a weak localization of solutions for an arbitrary potential which degenerates only on the initial plane.Bibliography: 12 titles.

An O(k&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;+kh&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;+h&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;) Accurate Two-level Implicit Cubic Spline Method for One Space Dimensional Quasi-linear Parabolic Equations

In this piece of work, using three spatial grid points, we discuss a new two-level implicit cubic spline method of O(k2 + kh2 + h4) for the solution of quasi-linear parabolic equation , 0 0 subject to appropriate initial and Dirichlet boundary conditions, where h > 0, k > 0 are grid sizes in space and time-directions, respectively. The cubic spline approximation produces at each time level a spline function which may be used to obtain the solution at any point in the range of the space variable. The proposed cubic spline method is applicable to parabolic equations having singularity. The stability analysis for diffusion- convection equation shows the unconditionally stable character of the cubic spline method. The numerical tests are performed and comparative results are provided to illustrate the usefulness of the proposed method.

Optimal regularity estimates for parabolic p-harmonic equations

In this paper optimal regularity estimates for weak solutions of quasilinear parabolic equations of p-Laplacian type with small BMO coefficients are investigated. Our results improve the known results for such equations using a harmonic analysis free technique.

Lipschitz bounds for solutions of quasilinear parabolic equations in one space variable

We bound the modulus of continuity of solutions to quasilinear parabolic equations in one space variable in terms of the initial modulus of continuity and elapsed time. In particular we characterize those equations for which the Lipschitz constants of solutions can be bounded in terms of their initial oscillation and elapsed time.