The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blömker, M. Hairer, G.A. Pavliotis, Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 7) (2007) 1721 1744].
The multivalued term in problem (1.1) that we consider contains not only the gradient but also a Laplacian item, and we obtain results in a weak solution sense.
As in many other cases the existence or nonexistence of positive solutions for (1.1) is closely related to the existence or nonexistence of positive solutions of the corresponding the steady-state problems, so that we consider first the following nonlinear elliptic boundary value problem: textstylebegin{cases} -mathcal{A}u=f x,u,K*u) &text{in } Omega, mathcal{B}u=g(x) &text{on } partialOmega.
When climate scientists started to investigate the impacts of climate change, they did what scientists have always done: we sought the hardest problems that we considered solvable and went for them.
This means that we consider the problem (221).
Let be a positive integer and a -strict pseudo-contraction for some such that We consider the problem of finding such that (4.1).
We will say that a certain property holds in ] α, β [ if it takes place on every closed subinterval of ] α, β [. Recall that we consider the problem (1), (2), where p, q ∈ L loc ( ] a, b [ ). Theorem 1.2 Let (3) hold. Then the homogeneous problem (1a), (2) has no more than one, up to a constant multiple, nontrivial solution.
Now we state the problem that we will consider along the paper: Problem B To find a solution of Eq. (1), satisfying boundary conditions (3).
