In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation

The aim of this paper is to establish the existence theory to nonlinear parabolic equations with nonstandard

Moreover, we will show that two unique weak solutions

The motivation of this paper contains several aspects. The first one is that in general parabolic problems are important for the modelling of space- and time-dependent problems, e.g., problems from physics or biology. In particular, evolutionary equations and systems can be used to model physical processes, e.g., heat conduction or diffusion processes. One example is the Navier–Stokes equation, the basic equation in fluid mechanics. Furthermore, we want to refer to [

The second interesting aspect of this paper is the nonstandard growth setting. Such setting arises for example by studying certain classes of non-Newtonian fluids such as electro-rheological fluids or fluids with viscosity depending on the temperature. Some properties of solutions to systems of such modified Navier–Stokes equation are studied in [

In the context of parabolic problems with

The third interesting aspect of the investigation of problems related to (

For this equation, the first eigenvalue is the minimum of the Rayleigh quotient

In this paper, we consider a bounded domain

Moreover, the parabolic distance is given by

Finally, we point out that the monotonicity condition (5) implies, by using the growth condition (

The spaces

The set

Notice that we will use also the abbreviation

In addition, we define

Furthermore, we consider more general nonstandard Sobolev spaces without fixed

If

Our next aim is to introduce the dual space of

Notice that whenever (

Finally, we are in the situation to give the definition of a weak solution to the parabolic nonstandard growth equation (

In the following, we mention our main result and we briefly describe the strategy of the proof to these results and the novelties of the paper. We start with some useful and important preliminary results (see

In the following, we will refer to some useful tools, which we will need for our proof. First of all, we refer to two lemmas, which are useful tools when dealing with p-growth problems. To this aim, we define a function

Since

Finally, we need the following Theorem ([

First of all, we will prove the existence of a unique weak solution to the Dirichlet problem (

We start by constructing a sequence of the Galerkin’s approximations, where the limit of this sequence is equal to the solution in (

Then, Equation (

Please notice that, if

Next, we want to derive a uniform bound for

Then, we choose a test function

Note that

Then, we derive by utilizing the growth condition (

Therefore, we have a uniform bound of

Moreover, by Theorem 2, we can conclude that the sequence

Furthermore, the growth assumption of

Our next aim is to show that

Moreover, it follows from (

Adding (

Then, we test Equation (

Then, passing to the limit

Moreover, we have to show that

Furthermore,

Next, we prove the uniqueness of the weak solution. Therefore, we assume that there exist two weak solutions

Using the monotonicity condition (5) and Lemma 2, we derive at

Therefore, we have that

Finally, we prove the stability of the weak solution to the Dirichlet problem (

Using the monotonicity condition (5) and Lemma 2, we derive at

In this manuscript we proved the existence of a unique weak solution to the Dirichlet problem (

The author wishes to thank the referees for their careful reading of the original manuscript and their comments that eventually led to an improved presentation.

The author declares no conflict of interest.

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