^{1, 2}

^{2}

^{2}

^{1}

^{2}

The existence and uniqueness of global mild solutions are proven for a class of semilinear nonautonomous evolution equations. Moreover, it is shown that the system, under
considerations, has a unique steady state. This analysis uses,
essentially, the dissipativity, a subtangential condition, and the
positivity of the related

Several chemical and biochemical processes are typically described by nonlinear coupled
partial differential equations “PDE” and hence by distributed
parameter models (see [

Additionally, the existence and uniqueness of the corresponding equilibrium profile will be proven.

In the above equations,

(i) The nonlinear
models considered in this paper have been studied in a qualitative manner by
several authors. In the case,

In practice, the special cases

Recently, the existence of global solutions for
problems such as (

(ii) For technological limitations and economical considerations, the following
saturation conditions are usually fulfilled for all

This paper is organized as follows. In Section

Let

This section is devoted to investigate sufficient
conditions for the existence and uniqueness of global mild solutions to the
following abstract Cauchy problem:

The semilinear nonautonomous evolution equations have
been treated by a number of authors [

The following result gives sufficient conditions for
the existence and uniqueness of global mild solutions to the semilinear
equations of type (

Suppose that the following conditions are fulfilled:

It is shown in [

In the particular case when

Let

Let

Theorem

Suppose that the following conditions are
fulfilled:

Throughout the
sequel, we assume

Clearly, the Hilbert space

In the following, we will assume that

This PDEs describing the reactor dynamics may be
formally written in the abstract form as

The linear operator

This section is concerned with the existence and the
uniqueness of mild solution for our problem given by (

For each

Let

Let, now,

The following lemma is useful to establish the dissipativity property.

There exists

Let

Finally, we state the invariance properties of the state trajectories of the model given by (

One has that

Let

Now, we are in
a position to state and prove our global existence result for problem (

Let

Since

Let

Let

Thus the proof of the theorem is complete.

The next section deals with the existence and uniqueness results of equilibrium profile solutions for a nonlinear model given by
(

In the steady-state solution analysis, the inlet functions

The tubular reactor modelled by the nonlinear coupled partial differential equations given
by (

The sequel of this paper will deal with the uniqueness
analysis of steady states in the important case where

First, since

Now, we derive a positivity lemma, which will play a fundamental role in the proof of the uniqueness result of steady states.

Let

Let

For

Let

By similar considerations as above, we also get

Now, we have the following system:

Multiplying
(

Let

In this paper, we have studied the existence and uniqueness of the global mild solution for a
class of tubular reactor nonlinear nonautonomous models. It has also been
proven that the trajectories are satisfying time-dependent constraints, that
is,

An important open question is the stability analysis
of equilibrium profile for system (

This paper presents research results of the Moroccan “Programme Thématique d'Appui à la Recherche Scientifique”