scholarly journals Hölder regularity for degenerate parabolic equations with variable exponents

2022 ◽  
Vol 40 ◽  
pp. 1-19
Author(s):  
Hamid EL Bahja
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equations ◽  
Hölder Regularity ◽  
Variable Exponents ◽  
Young’S Inequality ◽  
Scaling Method ◽  
Degenerate Parabolic ◽  
Young's Inequality ◽  
Holder Regularity

In this paper, we discuss a class of degenerate parabolic equations with variable exponents. By  using the Steklov average and Young's inequality, we establish energy and logarithmicestimates for solutions to these equations. Then based on the intrinsic scaling method, we provethat local weak solutions are locally continuous.

scholarly journals Hölder continuity of singular parabolic equations with variable nonlinearity

2020 ◽  
Vol 28 (3) ◽  
pp. 51-82
Author(s):  
Hamid El Bahja
Keyword(s):  
Iterative Method ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Hölder Continuity ◽  
Regularity Result ◽  
Hölder Regularity ◽  
Variable Exponents ◽  
Holder Continuity ◽  
Singular Parabolic Equations ◽  
Holder Regularity

AbstractIn this paper we obtain the local Hölder regularity of the weak solutions for singular parabolic equations with variable exponents. The proof is based on DiBenedetto’s technique called intrinsic scaling; by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result.

scholarly journals Decreasing of the L^1 norm and mass conservation for Porous Medium Equations with advection

2018 ◽  
Vol 4 (2) ◽  
pp. 67-77
Author(s):  
Nicolau Matiel Lunardi Diehl ◽  
Lucinéia Fabris
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Weak Solutions ◽  
Parabolic Equations ◽  
Mass Conservation ◽  
Degenerate Parabolic Equations ◽  
Conservation Property ◽  
The Cauchy Problem ◽  
Porous Medium Equations ◽  
Degenerate Parabolic

In this paper, we show that the $L^1$ norm of the bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the formu_t + div f(x,t,u) = div(|u|^{\alpha}\nabla u),   x \in R^n , t > 0,where \alpha > 0 is constant, decrease, under fairly broad conditions in advection flow f. In addition, we derive the mass conservation property for positive (or negative) solutions.

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