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

2021/02/02 by Igor I. Skrypnik, Skrypnik, Igor I., Mykhailo V. Voitovych +1
Mathematics · #35B65 #35D30 #35K59 #35K92 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2102.01550

openalex publication_date 2021/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the continuity of bounded solutions for a wide class of parabolic\nequations with (p,q)-growth u_t-
rm div
left(g(x,t,|
nablaν|)
,
frac
nabla u|
nabla u|
right)=0, under the generalized\nnon-logarithmic Zhikov's condition g(x,t,
rm v/r)
leqslant\nc(K)
,g(y,
tau,
rm v/r),
quad (x,t), (y,
tau)
in Q_r,r(x₀,t₀),
quad\n0lt;
rm v
leqslant K
lambda(r),
quad\n
lim
limits_{r
rightarrow0}
lambda(r)=0,
quad
lim
limits_{r
rightarrow0}\n
frac
lambda(r)r=+
infty,
quad
int₀
lambda(r)
,
fracdrr=+
infty.\n In particular, our results cover new cases of double-phase parabolic\nequations.\n

Related

• Continuity at a boundary point of solutions to quasilinear elliptic equations with generalized Orlicz growth and non-logarithmic conditions
• B₁ classes of DeGiorgi-Ladyzhenskaya-Ural'tseva and their applications to elliptic and parabolic equations with generalized Orlicz growth conditions
• Harnack's inequality for degenerate double phase parabolic equations under the non-logarithmic Zhikov's condition
• Quantitative estimates of L^p maximal regularity for nonautonomous operators and global existence for quasilinear equations
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