The Cauchy Problem for Parabolic Equations with Degeneration

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...

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...

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

Modified Decomposition Method for Solving the Cauchy Problem for Nonlinear Parabolic-Hyperbolic Equations

In this paper Modified decomposition method is applied to the solvability of nonlinear parabolic-hyperbolic equations and illustrated with a few simple examples.

THE CAUCHY PROBLEM FOR p-EVOLUTION EQUATIONS

In this paper we deal with the Cauchy problem for evolution equations with real characteristics. We show that the problem is well-posed in Sobolev spaces assuming a suitable decay of the coefficients as the space variable x → ∞. In some cases, such a decay may also compensate a lack of regularity with respect to the time variable t.