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Register a new userAsymptotic behavior in time periodic parabolic problems with unbounded coefficients
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We study asymptotic behavior in a class of non-autonomous second order parabolic equations with time periodic unbounded coefficients in $mathbb Rtimes mathbb R^d$. Our results generalize and improve asymptotic behavior results for Markov semigroups having an invariant measure. We also study spectral properties of the realization of the parabolic operator $umapsto {cal A}(t) u - u_t$ in suitable $L^p$ spaces.
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We study a class of elliptic operators $A$ with unbounded coefficients defined in $ItimesCR^d$ for some unbounded interval $IsubsetCR$. We prove that, for any $sin I$, the Cauchy problem $u(s,cdot)=fin C_b(CR^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family ${G(t,s)}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for ${G(t,s)}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures.
We are considering the asimptotic behavior as $ttoinfty$ of solutions of the Cauchy problem for parabolic second order equations with time periodic coefficients. The problem is reduced to considering degenerate time-homogeneous diffusion processes on the product of a unit circle and Euclidean space.
We consider Cauchy problem for a divergence form second order parabolic operator with rapidly oscillating coefficients that are periodic in spatial variable and random stationary ergodic in time. As was proved in [25] and [13] in this case the homogenized operator is deterministic. We obtain the leading terms of the asymptotic expansion of the solution , these terms being deterministic functions, and show that a properly renormalized difference between the solution and the said leading terms converges to a solution of some SPDE.
We establish the $L_p$-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [6] to a large extent.
We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.
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