 Abstract | This thesis is concerned with stochastic heat equation with memory and nonlinear
energy supply. The main motivation to study such systems comes from Thermodynamics,
see [85]. The main objective of this work is to study the existence and
uniqueness of solutions to such equations and to investigate some fundamental
properties of solutions like continuous dependence on initial conditions. In our
approach we follow the seminal papers by Da Prato and Clement [10], where the
stochastic heat equation with memory is tranformed into an integral equation in
a function space and the so-called mild solutions are studied. In the aforementioned
papers only linear equations with additive noise were investigated. The main contribution of this work is the extension of this approach to nonlinear equations.
Our main tools are the theory of stochastic convolutions as developed in
[33] and the theory of resolvent kernels for deterministic linear heat equations with
memory, see[10]. Since the solution at time t depends on the whole history of the
process up to time t, the resolvent kernel does not define a semigroup of operators
in the state space of the process and therefore a ”standard” theory of stochastic
evolution equations as presented in the monograph [33] does not apply. A more
delicate analysis of the resolvent kernles and the associated stochastic convolutions
is needed.
We will describe now content of this thesis in more detail. Introductory Chapters
1 and 2 collect some basic and essentially well known facts about the Wiener
process, stochastic integrals, stochastic convolutions and integral kernels. However,
some results in Chapter 2 dealing with stochastic convolution with respect to
non-homogenous Wiener process are extensions of the existing theory. The main
results of this thesis are presented in Chapters 3 and 4. In Chapter 3 we prove
the existence and uniqueness of solutions to heat equations with additive noise
and either Lipschitz or dissipative nonlinearities. In both cases we prove the continuous
dependence of solutions on initial conditions. In Chapter 4 we prove the
existence and uniqueness of solutions and continuous dependence on initial conditions
for equations with multiplicative noise. The diffusion coefficients defined by
unbounded operators are allowed. |
