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T:A:L:K:S
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title:
On a p(t,\omega,x)-Laplace evolution equation with a stochastic force |
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name:
Zimmermann |
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first name:
Aleksandra
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location/conference:
RDSN14
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abstract:
We are interested in the existence and uniqueness of the following nonlinear parabolic problem (P) of p(t,x,\omega)-Laplace
type:
du-div (|Du|^{p(t,\omega,x)-2}Du)dt=h(.,u)dW in \Omega\times (0,T)\times D
u=0 in \Omega\times(0,T) \partial D
u(0,.)=u_0 in L^2(D)
When p is a fixed exponent, the problem is a classical one and can be solved by monotonicity arguments and Minty's trick comes from the possibility to write an Itô Formula in this setting. Since the Lebesgue and Sobolev spaces with variable exponent of variables t, \omega and x are Orlicz type spaces and do not fit into the classical framework of Bochner spaces, the arguments have to be adapted to the new setting. In particular, a proof of the Itô formula by a time discretization is out of range and since the variable exponent Lebesgue spaces are not stable by partial convolution in only the variable x, a proof of Itô formula using convolution fails as well. Our steps are the following ones: First, we consider a singular perturbation of our problem (P) with a 'nice' function h independent of u and we obtain a stability result of the solution with respect to h. Passing to the limit with respect to the singular perturbation we prove that the problem (P) is well posed for an additive, 'nice' noise fuction h. Then we prove the result for any h \in N^2_W(0,T;L^2(D)) by a density argument. In the last step we solve problem (P) for a multiplicative noise by a fixed-point argument. |
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