UNIQUENESS IN THE CAUCHY PROBLEM FOR PARABOLIC EQUATIONS

Existence and Uniqueness to the Cauchy Problem for Linear and Semilinear Parabolic Equations with Local Conditions

We consider the Cauchy problem in R for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The line...

The Cauchy Problem for Semilinear Parabolic Equations in Besov Spaces

In this paper we first give a unified method by introducing the concept of admissible triplets to study local and global Cauchy problems for semi-linear parabolic equations with a general nonlinear term in different Sobolev spaces. In particular, we establish the local well-posedness and small global well-posedness of the Cauchy problem for semi-linear parabolic equations without the homogeneou...

Backward Uniqueness for Parabolic Equations

It is shown that a function u satisfying |∂t + u| M (|u| + |∇u|), |u(x, t)| MeM|x| in (R \ BR) × [0, T ] and u(x, 0) = 0 for x ∈ R \ BR must vanish identically in R \ BR × [0, T ].

Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations

Attractors for Nonautonomous Parabolic Equations without Uniqueness

Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connec...