Abstract We provide necessary conditions on Euclidean domains for inclusions $$W^{1,p(\cdot )}(\Omega ) \hookrightarrow L^{q(\cdot $$ W 1 , p ( · ) Ω <mml:mo...

In this paper, we discuss a class of quasilinear evolution variational inequalities with variable exponent growth conditions in a generalized Sobolev space. We obtain the existence of weak solutions by means of penalty method. Moreover, we study the extinction properties of weak solutions to parabolic inequalities and provide a sufficient condition that makes the weak solutions vanish in a fini...

We investigate the problem of approximating function f in power-type weighted variable exponent Sobolev space Wr,p(.) ?(.) (0,1), (r = 1, 2, ...), by Hardy averaging operator A (f) (x) 1/x ?x0 f(t)dt. If lies ?(.)(0, 1), it is shown that A||(f)?f|| p(.),?(.)?rp(.) ? C ||f(r) p(.),?(.) , where a positive constant. Moreover, we consider boundedness grand Lebesgue spaces Lp(.),? ?(.)(0,1). The suf...

Since the space L x and W1,p x were thoroughly studied by Kováčik and Rákosnı́k 1 , variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In 2 , Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids. In recent years, the differential equations and variational problems with p x -growth cond...