Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFundamental solutions for second order parabolic systems with drift terms
75
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We construct fundamental solutions of second-order parabolic systems of divergence form with bounded and measurable leading coefficients and divergence free first-order coefficients in the class of $BMO^{-1}_x$, under the assumption that weak solutions of the system satisfy a certain local boundedness estimate. We also establish Gaussian upper bound for such fundamental solutions under the same conditions.
rate research
Read More
Auscher, McIntosh and Tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on $mathbb{R}^n$. In particular, in the case when $n=2$ they obtained Gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients. We study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded, measurable, time-independent coefficients, and extend their results to the systems of parabolic equations.
We construct Greens functions for second order parabolic operators of the form $Pu=partial_t u-{rm div}({bf A} abla u+ boldsymbol{b}u)+ boldsymbol{c} cdot abla u+du$ in $(-infty, infty) times Omega$, where $Omega$ is an open connected set in $mathbb{R}^n$. It is not necessary that $Omega$ to be bounded and $Omega = mathbb{R}^n$ is not excluded. We assume that the leading coefficients $bf A$ are bounded and measurable and the lower order coefficients $boldsymbol{b}$, $boldsymbol{c}$, and $d$ belong to critical mixed norm Lebesgue spaces and satisfy the conditions $d-{rm div} boldsymbol{b} ge 0$ and ${rm div}(boldsymbol{b}-boldsymbol{c}) ge 0$. We show that the Greens function has the Gaussian bound in the entire $(-infty, infty) times Omega$.
We establish existence and various estimates of fundamental matrices and Greens matrices for divergence form, second order strongly parabolic systems in arbitrary cylindrical domains under the assumption that solutions of the systems satisfy an interior H{o}lder continuity estimate. We present a unified approach valid for both the scalar and the vectorial cases.
We consider a second-order parabolic equation in $bR^{d+1}$ with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Holder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the $L_{infty}$-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.
We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
