scholarly journals On Solutions of a Parabolic Equation with Nonstandard Growth Condition

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Growth Condition ◽  
Boundary Value ◽  
Growth Conditions ◽  
Boundary Value Condition ◽  
Nonstandard Growth ◽  
Partial Boundary Value Condition ◽  
Nonstandard Growth Condition ◽  
Nonstandard Growth Conditions ◽  
The Stability

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient bx,t on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω×0,T. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.

The Dirichlet Problem for Harmonic Functions in the Smirnov Class with Variable Exponent

2007 ◽  
Vol 14 (2) ◽  
pp. 289-299
Author(s):  
Vakhtang Kokilashvili ◽  
Vakhtang Paatashvili
Keyword(s):  
Dirichlet Problem ◽  
Explicit Form ◽  
Growth Condition ◽  
Harmonic Functions ◽  
Variable Exponent ◽  
Domain Boundary ◽  
Problem Solution ◽  
Smirnov Class ◽  
Nonstandard Growth ◽  
Nonstandard Growth Condition

Abstract A solution of the Dirichlet problem for harmonic functions from the Smirnov class is obtained in the framework of functional spaces with a nonstandard growth condition. It is found that the domain boundary geometry influences the character of a problem solution. In the case of solvability, solutions are constructed in explicit form.

scholarly journals On compact and bounded embedding in variable exponent Sobolev spaces and its applications

2019 ◽  
Vol 9 (2) ◽  
pp. 401-414
Author(s):  
Farman Mamedov ◽  
Sayali Mammadli ◽  
Yashar Shukurov
Keyword(s):  
Sobolev Space ◽  
Growth Condition ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Lebesgue Spaces ◽  
Nonstandard Growth ◽  
Variable Exponent Sobolev Spaces ◽  
Nonstandard Growth Condition ◽  
Nonlinear Ode ◽  
Weighted Variable

Abstract For a weighted variable exponent Sobolev space, the compact and bounded embedding results are proved. For that, new boundedness and compact action properties are established for Hardy’s operator and its conjugate in weighted variable exponent Lebesgue spaces. Furthermore, the obtained results are applied to the existence of positive eigenfunctions for a concrete class of nonlinear ode with nonstandard growth condition.

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