Nonlinear parabolic problems in Musielak--Orlicz spaces

By Agnieszka Świerczewska-Gwiazda. 2013

52 citations recorded by the source corpus.

Our studies are directed to the existence of weak solutions to a parabolic problem containing a multi-valued term. The problem is formulated in the language of maximal monotone graphs. We assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of time and space dependent $N$--function. This results in formulation of the problem in generalized Musielak-Orlicz spaces. We are using density arguments, hence an important step of the proof is a uniform boundedness of appropriate convolution operators in Musielak-Orlicz spaces. For this purpose we shall need to assume a kind of logarithmic H\"older regularity with respect to $t$ and $x$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de0907e68cfd359624350574fd05d6e8c7fee71b

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Citations in the corpus

New martingale inequalities and applications to Fourier analysis (2018, background): “The Musielak–Orlicz spaces not only have their own interest, but they are also very useful in partial differential equations [2, 5, 28, 26], in calculus of variations [13], in image restoration [27, 43] and in fluid dynamics [73, 51].”
Extrapolation and interpolation in generalized Orlicz spaces (2018, background): “They have appeared in many problems in PDEs and the calculus of variations [4, 5, 6, 13, 27] and have applications to image processing [10, 32, 37] and fluid dynamics [53].”
Generalized Orlicz spaces and related PDE (2016, background): “In addition to being a natural generalization which covers r sults from both variable exponent and Orlicz spaces, the study of generalized Orlicz spaces can be motivated by applications to image processing [2, 6, 16], fluid dynamic s [33, 34] and differential equations.”
Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces (2015, background): “A generalization of log-Hölder condition of exponents for M(x, ξ) was assumed in [44] and [45], whereas the first paper concerns isotropic (t, x)−dependent N function and the second one x−dependent anisotropic N -function.”
Gossez's approximation theorems in Musielak–Orlicz–Sobolev spaces (2017, background): “Note that the results in [5, 6, 38] do not cover the functions e−1 and t/p(x) and the two phase function s+a(x)s unless p and a are constant functions, which are not excluded in our setting.”
Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space (2017, unknown): “The improvement of this result for the vector Orlicz spaces was given in [21], while for the x–dependent log-Hölder continuous modular functions in [7], developed in [19, 40] and further in [41] in the case of log-Hölder continuous modular functions dependent on x, as well as on t.”
Hölder regularity of quasiminimizers under generalized growth conditions (2017, background): “In addition to being a natural generalization which covers results from both variable exponent and Orlicz spaces, the study of nonstandard growth problems can be motivated by applications to image processing [1, 6, 21], fluid dynamics [40, 41] and differential equations [2, 14, 17, 18, 37, 38] The s”
A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces (2018, background): “We refer also to [107, 108, 110, 111, 196, 197] for some developments arising around the theory of non-Newtonian fluids and for existence to some parabolic problems within the setting to [183, 184].”