Multiplicity of solutions for a higher <inline-formula><tex-math id="M1">$ m $</tex-math></inline-formula>-polyharmonic Kirchhoff type equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Ampere type equation on unbounded domains. For a few special kinds of unbounded convex domains,we find the explicit formulas of the solutions to the problem. For general unbounded convex domain $\Om$, we prove the existence for solutions to the problem in the space$C^{\infty}(\Om)\cap~C~(\overline{\Om})$. We also obtain the local $C^{\frac{1}{2}}$-estimate up to the $\partial~\Omega$ and the estimate for the lower bound of the solutions.

Classical solvability is established for a certain nonlinear integrodifferential parabolic equation, on unbounded domains in several dimensions. The model equation of the Fokker-Planck type represents a regularized version of an equation recently derived by J. A. Acebron and R. Spigler for the physical problem of describing the time evolution of large populations of nonlinearly globally coupled random oscillators. Precise estimates are obtained for the decay of convolutions with fundamental solutions of linear parabolic equations on unbounded domains in R n . Existence of a classical solution with special properties is established.

ABSTRACTIn this paper, we extend some results to anisotropic variable exponent Sobolev spaces in unbounded domains. Building on the presented results, we prove the existence of at least three weak solutions for an anisotropic Schrödinger–Kirchhoff‐type potential system involving variable exponents. Subject to appropriate conditions on the nonlinearities, we obtain the existence of a minimum of three weak solutions for our problem.

This paper is concerned with the pullback attractors for the Kirchhoff type BBM equations defined on unbounded domains. Sobolev embeddings are invalid on unbounded domains. We obtain the pullback asymptotic compactness of such non-autonomous BBM equations by using the method of uniform tail-estimates.

We consider the first boundary-value problem for a third-order equation of combined type. Using the Saint-Venant principle, we study the uniqueness class for solutions of the problem in an unbounded domain.

We study the unique solvability of a problem with shift for an equation of mixed type in an unbounded domain. We prove the uniqueness theorem under inequality-type constraints for known functions for various orders of the fractional differentiation operators in the boundary condition. The existence of a solution is proved by reduction to a Fredholm equation of the second kind, whose unconditional solvability follows from the uniqueness of the solution of the problem.

ABSTRACTIn this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations of Kirchhoff type First, we prove the well-posedness of solution for the nonclassical diffusion equations of Kirchhoff type with critical nonlinearity on , then the existence of global attractor is established in the natural energy space . Finally, we obtain the existence of exponential attractors in , as a product, the global attractor has finite fractal dimension.

In this work, in an unbounded domain, we prove the cof the problem with combined Tricomi and Frankl conditions on one boundary characteristic for one class of equations of mixed type.

We consider a boundary value problem for an equation of the mixed type with a singular coefficient in an unbounded domain. The uniqueness of the solution of the problem is proved with the use of the extremum principle. In the proof of the existence of a solution of the problem, we use the method of integral equations.

Spectral method for differential equations of degenerate type on unbounded domains by using generalized Laguerre functions

This paper provides an existence theorem for the Dirichlet problem for a quasilinear partial differential equation of elliptic type in an unbounded domain. The principal new feature of this work is that the results are obtained under weaker monotonicity conditions on the coefficients of the equation than those employed in earlier work in this area.

This work is devoted to study the existence of solutions to equations of the p Laplacian type in unbounded domains. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. We apply the mountain pass theorem in weighted Sobolev spaces.

We consider equations describing the thermoelastic behavior of plates modeled in the Green-Lindsay sense. This is done with two different type of couplings of the fourth-order plate Kirchhoff-type plate equation to a second-order heat equation of Cattaneo type, once of second, and once of first order. We investigate both systems for bounded domains and for the Cauchy problem, asking for exponential stability in bounded domains resp. polynomial decay rates for the Cauchy problem. It turns out that one system is exponentially stable, while the other is not, and that, in correspondence, one does not have and the other one has regularity loss in the Cauchy problem. This provides a new interesting example where the different couplings lead to qualitatively different behavior, as known before for classical thermoelastic plates, for Timoshenko systems, for porous elasticity or for plates with two temperatures, with Fourier resp. Cattaneo heat conduction. The optimality of the decay rates obtained is also proved.

We consider the global existence and asymptotic stability of solutions to the Cauchy problem for degenerate nonlinear wave equations of Kirchhoff type with a dissipative term in unbounded domain. We derive the sharp decay estimates of the solution and its derivatives. Moreover, we show that the solution has a lower decay estimate of some algebraic rate.

Consider the Cauchy problem in unbounded domain for the degenerate wave equation of Kirchhoff type with damping: We show the existence of a unique global C2-solution in some Sobelev space, and derive the algebraic decay rates for the solution and its energy of the degenerate problem.