scholarly journals Existence and uniqueness for doubly nonlinear parabolic equations with nonstandard growth conditions

2012 ◽  
pp. 67-94 ◽  
Author(s):  
Stanislav Antontsev ◽  
Sergey Shmarev
Keyword(s):  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlinear Parabolic Equations ◽  
Growth Conditions ◽  
Nonstandard Growth ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Nonstandard Growth Conditions

scholarly journals Harnack’s inequality for doubly nonlinear equations of slow diffusion type

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Verena Bögelein ◽  
Andreas Heran ◽  
Leah Schätzler ◽  
Thomas Singer
Keyword(s):  
Vector Field ◽  
Harnack Inequality ◽  
Parabolic Equations ◽  
Full Range ◽  
Nonlinear Parabolic Equations ◽  
Growth Conditions ◽  
Diffusion Type ◽  
Slow Diffusion ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

scholarly journals Local subestimates of solutions to double-phase parabolic equations via nonlinear parabolic potentials

2019 ◽  
Vol 16 (1) ◽  
pp. 28-45
Author(s):  
Kateryna Buryachenko
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Growth Conditions ◽  
Local Boundedness ◽  
Nonstandard Growth ◽  
Right Hand ◽  
Double Phase ◽  
Nonlinear Parabolic ◽  
Nonstandard Growth Conditions ◽  
The Right

For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation.

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