arXiv subjects

Sergey Shmarev

Publications and source records attributed to Sergey Shmarev.

9 recordsLinked to original sources

arXiv2026

Global Calderon-Zygmund estimates for irregular double-phase evolution problem with non-divergence data

We study irregular double-phase parabolic equations with variable exponents and non-divergence data, \[ u_t-\operatorname{div} \left(\mathcal{F}(z,\nabla u)\nabla u \right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T), \] under the homogeneous Dirichlet boundary conditions. Here, $\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain, $T>0$, \[ \mathcal{F}(z,\nabla u)=a(z)|\nabla u|^{p(z)-2} + b(z) |\nabla u |^{q(z)-2} \] with given Lipschitz-continuous exponents $p,q$ that satisfy a suitable balance condition. The nonnegative coefficients $a(z), b(z)$ satisfy the inequality $a(z)+b(z)>0$ in $Q_T$, the space and time derivatives of $a$ and $b$ belong to $L^d(Q_T)$ with some $d$ depending on the data. If \[ f\in L^\sigma(Q_T) \quad \text{for} \ \sigma \in (2, N+2] \quad \text{and} \quad \mathcal{F}((\cdot,0),\nabla u_0)\,|\nabla u_0|^{r+2}\in L^1(\Omega), \] where $0\le r\le K(N,\sigma,p,q)$ if $\sigma<N+2$, while $r\ge0$ is arbitrary if $\sigma=N+2$, then the problem has a unique strong solution, for which we prove the global transfer of integrability from the initial data and the forcing term to the double-phase flux in the spirit of Calder\'on-Zygmund theory, higher integrability of the gradient, and the second-order space regularity: \[ \begin{split} & \text{$\mathcal{F}((\cdot,t),\nabla u(\cdot,t))|\nabla u(\cdot,t)|^{r+2}\in L^1(\Omega)$ for a.e. $t\in (0,T)$}, \\ & \text{$|\nabla u|^{2(\min\{p(z),q(z)\}-1)+r+s}\in L^1(Q_T)$ for every $s\in\left(0,\frac{4}{N+2}\right)$}, \\ & \mathcal{F}(z,\nabla u)|\nabla u|^{\frac{r+2}{2}} \in L^2(0,T;W^{1,2}(\Omega)). \end{split} \] The results improve and complement the results in \cite{Arora-Shmarev-JGA-2026} and extend them to the full range $r \geq 0$.

math.AP↗

arXiv2025

Irregular double-phase evolution problem: existence and global regularity

We investigate the homogeneous Dirichlet problem for the irregular double-phase evolution equation \[ u_t-\operatorname{div} \left( a(z)|\nabla u|^{p(z)-2} \nabla u + b(z)|\nabla u|^{q(z)-2} \nabla u\right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T), \] where $\Omega \subset \mathbb{R}^N$, $N \geq 2$ is a bounded domain, $T>0$, The non-differentiable coefficients $a(z)$, $b(z)$, the free term $f$, and the variable exponents $p$, $q$ are given functions. The coefficients $a$ and $b$ are nonnegative, bounded, satisfy the inequality \[ a(z)+b(z)\geq \alpha \quad \text{in} \ Q_T, \quad \text{and} \quad |\nabla a|, |\nabla b|, a_t, b_t \in L^d(Q_T) \] for some constant $\alpha>0$, and with $d>2$ depending on $\sup p(z)$, $\sup q(z)$, $N$, and the regularity of initial data $u(x,0)$. The free term $f$ and initial data $u(x,0)$ satisfy \[ f\in L^\sigma(Q_T) \ \text{with} \ \sigma>2 \quad \text{and} \quad |\nabla u(x,0)|\in L^{r}(\Omega) \ \text{with} \ r\geq \max \bigg\{2,\sup_{Q_T}p(z),\sup_{Q_T}q(z)\bigg\}. \] The variable exponents $p,q \in C^{0,1}(\overline{Q}_T)$ satisfy the balance condition \[ \frac{2N}{N+2} < p(z), q(z)< +\infty \ \text{in} \ \overline Q_T \quad \text{and} \quad \max\limits_{\overline Q_T}|p(z)-q(z)|< \dfrac{2}{N+2}. \] Under the above assumptions, we establish the existence of a solution, which is obtained as the limit of classical solutions to a family of regularized problems and preserves initial temporal integrability: \[ |\nabla u(\cdot, t)| \in L^r(\Omega) \ \text{for a.e.} \ t \in (0,T), \] gains global higher integrability: \[ |\nabla u|^{\min\{p(z), q(z)\} + s +r} \in L^1(Q_T) \ \text{for any} \ s \in \left(0, \frac{4}{N+2}\right), \] and attains second-order regularity: \[ a(z) |\nabla u|^{\frac{p+r-2}{2}}+b(z) |\nabla u|^{\frac{q+r-2}{2}}\in L^2(0,T;W^{1,2}(\Omega)). \]

math.AP↗

arXiv2024

Global gradient estimates for solutions of parabolic equations with nonstandard growth

We study how the smoothness of the initial datum and the free term affect the global regularity properties of solutions to the Dirichlet problem for the class of parabolic equations of $p(x,t)$-Laplace type %with nonlinear sources depending on the solution and its gradient: \[ u_t-Δ_{p(\cdot)}u=f(z)+F(z,u,\nabla u),\quad z=(x,t)\in Q_T=Ω\times (0,T), \] with the nonlinear source $F(z,u,\nabla u)=a(z)|u|^{q(z)-2}u+|\nabla u|^{s(z)-2}(\vec c,\nabla u)$. It is proven the existence of a solution such that if $|\nabla u(x,0)|\in L^r(Ω)$ for some $r\geq \max\{2,\max p(z)\}$, then the gradient preserves the initial order of integrability in time, gains global higher integrability, and the solution acquires the second-order regularity in the following sense: \[ \text{$|\nabla u(x,t)|\in L^r(Ω)$ for a.e. $t \in (0,T)$}, \qquad \text{$|\nabla u|^{p(z)+ρ+r-2} \in L^1(Q_T)$ for any $ρ\in \left(0, \frac{4}{N+2}\right)$}, \] and \[ |\nabla u|^{\frac{p(z)+r}{2}-2}\nabla u\in L^2(0,T;W^{1,2}(Ω))^N. \] The exponent $r$ is arbitrary and independent of $p(z)$ if $f\in L^{N+2}(Q_T)$, while for $f\in L^σ(Q_T)$ with $σ\in (2,N+2)$ the exponent $r$ belongs to a bounded interval whose endpoints are defined by $\max p(z)$, $\min p(z)$, $N$, and $σ$. An integration by parts formula is also proven, which is of independent interest.

math.AP↗

arXiv2023

Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

We consider the homogeneous Dirichlet problem for the parabolic equation \[ u_t- \operatorname{div} \left(|\nabla u|^{p(x,t)-2} \nabla u\right)= f(x,t) + F(x,t, u, \nabla u) \] in the cylinder $Q_T:=Ω\times (0,T)$, where $Ω\subset \mathbb{R}^N$, $N\geq 2$, is a $C^{2}$-smooth or convex bounded domain. It is assumed that $p\in C^{0,1}(\overline{Q}_T)$ is a given function, and that the nonlinear source $F(x,t,s, ξ)$ has a proper power growth with respect to $s$ and $ξ$. It is shown that if $p(x,t)>\frac{2(N+1)}{N+2}$, $f\in L^2(Q_T)$, $|\nabla u_0|^{p(x,0)}\in L^1(Ω)$, then the problem has a solution $u\in C^0([0,T];L^2(Ω))$ with $|\nabla u|^{p(x,t)} \in L^{\infty}(0,T;L^1(Ω))$, $u_t\in L^2(Q_T)$, obtained as the limit of solutions to the regularized problems in the parabolic Hölder space. The solution possesses the following global regularity properties: \[ \begin{split} & |\nabla u|^{2(p(x,t)-1)+r}\in L^1(Q_T)\quad \text{for any $0 < r < \frac{4}{N+2}$}, \\ & |\nabla u|^{p(x,t)-2} \nabla u \in W^{1,2}(Q_T)^N. \end{split} \]

math.AP↗

arXiv2022

Existence and global second-order regularity for anisotropic parabolic equations with variable growth

We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinder $Ω\times (0,T)$, where $Ω\subset \mathbb{R}^N$, $N\geq 2$, is a parallelepiped. The exponents of nonlinearity $p_i$ are given Lipschitz-continuous functions. It is shown that if $p_i(x,t)>\frac{2N}{N+2}$, \[ μ=\sup_{Q_T}\dfrac{\max_i p_i(x,t)}{\min_i p_i(x,t)}<1+\dfrac{1}{N}, \quad |D_{x_i}u_0|^{\max\{p_i(\cdot,0),2\}}\in L^1(Ω),\quad f\in L^2(0,T;W^{1,2}_0(Ω)), \] then the problem has a unique solution $u\in C([0,T];L^2(Ω))$ with $|D_{x_i} u|^{p_i}\in L^{\infty}(0,T;L^1(Ω))$, $u_t\in L^2(Q_T)$. Moreover, \[ |D_{x_i}u|^{p_i+r}\in L^1(Q_T)\quad \text{with some $r=r(μ,N)>0$},\qquad |D_{x_i}u|^{\frac{p_i-2}{2}}D_{x_i}u\in W^{1,2}(Q_T). \] The assertions remain true for a smooth domain $Ω$ if $p_i=2$ on the lateral boundary of $Q_T$.

math.AP↗

arXiv2021

Existence and regularity results for a class of parabolic problems with double phase flux of variable growth

We study the homogeneous Dirichlet problem for the equation \[ u_t-\operatorname{div}\left((a(z)\vert \nabla u\vert ^{p(z)-2}+b(z)\vert \nabla u\vert ^{q(z)-2})\nabla u\right)=f\quad \text{in $Q_T=Ω\times (0,T)$}, \] where $Ω\subset \mathbb{R}^N$, $N\geq 2$, is a bounded domain with $\partialΩ\in C^2$. The variable exponents $p$, $q$ and the nonnegative modulating coefficients $a$, $b$ are given Lipschitz-continuous functions of the argument $z=(x,t)\in Q_T$. It is assumed that $\frac{2N}{N+2} 0$ in $\overline{Q}_T$},\; α=const;\qquad \text{$\vert p(z)-q(z)\vert <\frac{2}{N+2}$ in $\overline{Q}_T$}. \] We find conditions on the source $f$ and the initial data $u(\cdot,0)$ that guarantee the existence of a unique strong solution $u$ with $u_t\in L^2(Q_T)$ and $a\vert \nabla u\vert ^{p}+b\vert \nabla u\vert ^q\in L^\infty(0,T;L^1(Ω))$. The solution possesses the property of global higher integrability of the gradient, \[ \vert \nabla u\vert ^{\min\{p(z),q(z)\}+r}\in L^1(Q_T)\quad \text{with any $r\in \left(0,\frac{4}{N+2}\right)$}, \] which is derived with the help of new interpolation inequalities in the variable Sobolev spaces. The second-order differentiability of the strong solution is proven: \[ D_{x_i}\left(\left(a\vert \nabla u\vert ^{p-2}+b\vert \nabla u\vert ^{q-2}\right)^{\frac{1}{2}}D_{x_j}u\right)\in L^2(Q_T),\quad i,j=1,2,\ldots,N. \]

math.AP↗

arXiv2021

Strong solutions of the double phase parabolic equations with variable growth

This paper addresses the questions of existence and uniqueness of strong solutions to the homogeneous Dirichlet problem for the double phase equation with operators of variable growth: \[ u_t - div \left(|\nabla u|^{p(z)-2} \nabla u+ a(z) |\nabla u|^{q(z)-2} \nabla u \right) = F(z,u) \quad \text{in $Q_T=Ω\times (0,T)$} \] where $Ω\subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain with the boundary $\partialΩ\in C^2$, $z=(x,t)\in Q_T$, $a:\bar Q_T \mapsto \mathbb{R}$ is a given nonnegative coefficient, and the nonlinear source term has the form \[ F(z,v)=f_0(z)+b(z)|v|^{σ(z)-2}v. \] The variable exponents $p$, $q$, $σ$ are given functions defined on $\bar{Q}_T$, $p$, $q$ are Lipschitz-continuous and \[ \dfrac{2N}{N+2} 0. \]

math.AP↗

arXiv2021

Stability for evolution equations with variable growth

We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[ u_t-Δ_{p(x,t)}u=f(x,t,u),\quad (x,t)\in Q=Ω\times (0,T), \] where $Ω$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $p(x,t)$ is a given function $p(\cdot):Q\mapsto (\frac{2n}{n+2},p^+]$, $p^+<\infty$. It is shown that the solution is stable with respect to perturbations of the variable exponent $p(x,t)$, the nonlinear source term $f(x,t,u)$, and the initial data. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the nonlinearity exponent and the data $u(x,0)$, $f$. Estimates on the rate of convergence of a sequence of solutions to the solution of the limit problem are derived.

math.AP↗

arXiv2013

Existence and nonuniqueness of segregated solutions to a class of cross-diffusion systems

We study the the Dirichlet problem for the cross-diffusion system \[ \partial_tu_i=\operatorname{div}\left(a_iu_i\nabla (u_1+u_2)\right)+f_i(u_1,u_2),\quad i=1,2,\quad a_i=const>0, \] in the cylinder $Q=Ω\times (0,T]$. The functions $f_i$ are assumed to satisfy the conditions $f_1(0,r)=0$, $f_2(s,0)=0$, $f_1(0,r)$, $f_2(s,0)$ are locally Lipschitz-continuous. It is proved that for suitable initial data $u_0$, $v_0$ the system admits segregated solutions $(u_1,u_2)$ such that $u_i\in L^{\infty}(Q)$, $u_1+u_2\in C^{0}(\overline{Q})$, $u_1+u_2>0$ and $u_1\cdot u_2=0$ everywhere in $Q$. We show that the segregated solution is not unique and derive the equation of motion of the surface $Γ$ which separates the parts of $Q$ where $u_1>0$, or $u_2>0$. The equation of motion of $Γ$ is a modification of the Darcy law in filtration theory. Results of numerical simulation are presented.

math.AP↗