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1.
We consider initial boundary value problems for a third-order nonlinear pseudoparabolic equation with one space dimension. The boundary condition is given by an integral; the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet or Neumann counterparts. By means of appropriate elliptic estimates we are able to seek solutions not only in the weighted spaces but also in the usual Sobolev spaces. The procedure is carried out in a unified way. Our results characterize a regularity of the pseudoparabolic operator that is different from that of the parabolic operator.  相似文献   

2.
In this paper we prove existence and uniqueness of strong solutions to the homogeneous Neumann problem associated to a parabolic equation with linear growth with respect to the gradient variable. This equation is a generalization of the time-dependent minimal surface equation. Existence and regularity in time of the solution is proved by means of a suitable pseudoparabolic relaxed approximation of the equation and a passage to the limit.  相似文献   

3.
We study the existence and uniqueness of the solution of the inverse problem of finding an unknown coefficient b(x) multiplying the lower derivative in the nondivergence parabolic equation on the plane. The integral of the solution with respect to time with some given weight function is given as additional information. The coefficients of the equation depend on the time variable as well as the space variable.  相似文献   

4.
We consider the initial boundary value problem of the non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random noises. The space boundary is Lipschitz, and we impose nonzero condition on the parabolic boundary. We prove a regularity result by finding appropriate spaces for solutions and pre-assigned data in the problem. We use a collection of tools from potential theory, harmonic analysis, and probability. Some lemmas are as important as the main theorem.  相似文献   

5.
We consider linear parabolic equations of second order in a Sobolev space setting. We obtain existence and uniqueness results for such equations on a closed two-dimensional manifold, with minimal assumptions about the regularity of the coefficients of the elliptic operator. In particular, we derive a priori estimates relating the Sobolev regularity of the coefficients of the elliptic operator to that of the solution. The results obtained are used in conjunction with an iteration argument to yield existence results for quasilinear parabolic equations.  相似文献   

6.
A parabolic variational inequality is investigated which comes from the study of the optimal exercise strategy for the perpetual American executive stock options in financial markets. It is a degenerate parabolic variational inequality and its obstacle condition depends on the derivative of the solution with respect to the time variable. The method of discrete time approximation is used and the existence and regularity of the solution are established.  相似文献   

7.
The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some prior estimates under certain assumptions on the initial value of the equation. The local existence of its solution in Sobolev space Hs (R) with 1 〈 s ≤ 3/2 is derived.  相似文献   

8.
We consider some nonstandard Sobolev spaces in one dimension, in which functions have different regularity in different subsets. These spaces are useful in the study of some nonlinear parabolic equations where the nonlinearity is highly degenerate and depends on the smoothness of the solution at a certain subset (that may vary with time). An example of application is a diffusion equation with a smooth free boundary, and a moving source/sink where the solution has singularity. The main new idea here is to characterize the functional space setting that is needed for semigroup theory to apply.  相似文献   

9.
We consider a parabolic equation whose space operator is a product of a nonlinear bounded function which depends on a nonlocal characteristic with respect to a solution gradient and a strongly monotone potential operator. We prove the existence and uniqueness of the solution in the class of the vector-valued functions with values in the Sobolev space.  相似文献   

10.
We prove the optimal regularity, in Sobolev spaces, of the solution of a parabolic equation set in a triangular domain T. The right-hand term of the equation is taken in Lebesgue space Lp(T). The method of operators sums in the non-commutative case is referred to.  相似文献   

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