Abstract
We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.
Highlights
We study the existence and uniqueness of a classical solution to the CauchyDirichlet problem for a linear parabolic differential equation in a general unbounded domain
We prove the existence and uniqueness of a classical solution to 1.2, when the coefficients are locally Lipschitz continuous in x and locally Holder continuous in t, aij has a quadratic growth, bi has linear growth, and c is bounded from above
Observe that the local ellipticity is only assumed on 0, ∞ × D. This condition is used to prove the existence of a classical solution to 1.2 and so is only needed in that set
Summary
We study the existence and uniqueness of a classical solution to the CauchyDirichlet problem for a linear parabolic differential equation in a general unbounded domain. In the case of general unbounded domains, Fornaro et al in 18 studied the homogeneous, autonomous Cauchy-Dirichlet problem They proved, using analytical methods in semigroups, the existence and uniqueness of a solution to the Cauchy-Dirichlet problem when the coefficients are locally C1,α, with aij bounded, b and c functions with a Lyapunov type growth; that is, there exists a function φ ∈ C1,2 0, T × Rd such that lim inf φ t, x ∞. In Bertoldi et al generalized the method to nonconvex sets with C2 boundary They studied the existence, uniqueness, and gradient estimates for the Cauchy-Neumann problem.
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