The Noncharacteristic Cauchy Problem for Parabolic Equations in Two Space Variables

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...

Equations in Two Space Variables

Hölder estimates for spatial second derivatives are proved for solutions of fully nonlinear parabolic equations in two space variables. Related techniques extend the regularity theory for fully nonlinear parabolic equations in higher dimensions.

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.