Nonlinear diffusion problems featuring stochastic effects may be described by stochastic partial differential equations of the form [Formula: see text] We present an existence theory for such equations under general monotonicity assumptions on the nonlinearities. In particular, [Formula: see text], [Formula: see text], and [Formula: see text] are allowed to be multivalued, as required by the modelization of solid–liquid phase transitions. In this regard, the equation corresponds to a nonlinear-diffusion version of the classical two-phase Stefan problem with stochastic perturbation. The existence of martingale solutions is proved via regularization and passage-to-the-limit. The identification of the limit is obtained by a lower-semicontinuity argument based on a suitably generalized Itô’s formula. Under some more restrictive assumptions on the nonlinearities, existence and uniqueness of strong solutions follows. Besides the relation above, the theory covers equations with nonlocal terms as well as systems.
The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.
AbstractAn SEIR epidemic model with constant immigration and random fluctuation around the endemic equilibrium is considered. As a special case, a deterministic system discussed by Li et al. will be incorporated into the stochastic version given by us. We carry out a detailed analysis on the asymptotic behavior of the stochastic model, also regarding of the basic reproduction number ℛ
AbstractIn this paper, we are interested in $L^{\infty }$
L
∞
decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$
L
∞
decay estimates of weak solutiona.
AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$
∂
t
u
-
div
A
(
x
,
t
,
u
,
D
u
m
)
=
div
F
,
where the vector field $${\mathbf {A}}$$
A
fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$
m
>
0
and $$p>1$$
p
>
1
with $$m(p-1) > 1$$
m
(
p
-
1
)
>
1
are included in our considerations.
Doubly nonlinear stochastic evolution equations
