Unique Classical Solution Research Articles

Regularized semigroups of bounded semivariation

We use regularized semigroups to consider local linear and semilinear inhomogeneous abstract Cauchy problems on a Banach space in a unified way. We show that the inhomogeneous abstract Cauchy problem {fx43-1} has a unique classical solution, for allf eC([0,T], [Im(C)]),x inC(D(A)), if and only ifA generates aC-regularized semigroup of bounded semivariation, and has a strong solution for allf eL 1 ([0,T], [Im(C)]),x eC(D(A)) if and only if theC-regularized semigroup is what we call of bounded super semivariation. This includes locally Lipschitz continuousC-regularized semigroups. We give similar simple sufficient conditions for the semilinear abstract Cauchy problem {fx43-2} to have a unique solution.

Regular solutions to quasilinear integrodifferential equations in banach spaces

In this paper we establish the existence of a unique classical solution to a class of quasilinear integrodifferential equations in a general Banach space.

WELL-POSEDNESS OF A MIXED INITIAL–BOUNDARY VALUE PROBLEM FOR VIBRATING ELASTIC SYSTEMS

In this letter we consider the well-posedness of a mixed initial–boundary value problem for a vibrating elastic system, which describes the dynamics of a kind of flexible robot with an elastic link, a rigid link, and three rotation axes. The governing equations consist of two nonlinear partial differential equations of order four and three nonlinear integro-differential equations. Using the semigroup theory, we prove that the above-mentioned problem is well-posed, i.e. there is a unique classical solution that continuously depends on the initial data.

Quasilinear evolution equations with non-densely defined operators

Two problems for the abstract quasilinear evolution equation of "hyperbolic" type in a Banach space $$ u'(t) = A(t,u(t))u(t) \,\,\,\rm{for} \,\,\,t \geq 0, \,\,\, \rm{and} \,\,\, u(0)=u_0 $$ are studied without assuming that the domain of $A(t,w)$ is dense in the whole space. One is the fundamental problem of existence and uniqueness of classical solutions, and the other is the problem of extension or blow up of classical solutions.

Maximal regularity for a free boundary problem

This paper is concerned with the motion of an incompressible fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed, impermeable layer and above by a free surface moving under the influence of gravity. The laminar flow is governed by Darcy's law. We prove existence of a unique maximal classical solution, using methods from the theory of maximal regularity, analytic semigroups, and Fourier multipliers. Moreover, we describe a state space which can be considered as domain of parabolicity for the problem under consideration.

The problem of primary frost heave with frost penetration

AbstractThis paper is concerned with a mathematical model for the phenomenon of primary frost heave with frost penetration, which appears in the freezing process of wet soil. The problem is formulated in the form of a free boundary problem with two free boundaries. It is shown that there exists a unique classical solution for this problem.

On an initial boundary value problem for the equations of ideal magneto‐hydrodynamics

AbstractWe study an initial boundary value problem for the symmetric hyperbolic quasilinear system of the equations of ideal magneto‐hydrodynamics with a perfectly conducting wall boundary condition. Existence of a unique classical solution is proved inside a suitable class of functions of Sobolev type. Moreover, solutions inside the above class are shown to depend continuously in strong norm on the data.

Existence and uniqueness of solutions of the equations of motion for a fluid of second grade with non-homogeneous boundary conditions

In this paper we consider the equations of motion for a fluid of second grade together with non-homogeneous boundary conditions. We study the problem in a simply connected domain in R 3 and show the existence of a unique classical solution—local in time—under suitable regularity conditions on the boundary values. The method consists of an appropriate decomposition of the original problem along with application of multivalued fixed point theory.

Nonautonomous integro-differential equations of hyperbolic type

This paper is devoted to two problems for the nonautonomous integrodifferential equation in a Banach space $X$ of hyperbolic type $$ \text{$u'(t) = A(t)u(t)+ \int_0^t B(t,s)u(s) ds + f(t)$ for $t\in[0,T]$, and $u(0) = u_0$.} $$ One is the problem of existence and uniqueness of classical solutions without assuming that the common domain of $A(t)$ is dense in $X$, and the other is the regularity problem in the case where the common domain of $A(t)$ is dense in $X$. The regularity result will play a role in developing an abstract theory which can be applied to the second-order integrodifferential equations with the third kind boundary conditions.

Linear parabolic equations with venttsel initial boundary conditions

The Schauder estimates for solutions of linear second order parabolic equations with Venttsel initial boundary conditions are proved, and existence and uniqueness of classical solutions under such an initial boundary condition are established. An application to an engineering problem is also given.

Quenching for degenerate semilinear parabolic equations

Let q and a be nonzero constants, and for some constant c such that . We show existence of a unique classical solution for the degenerate parabolic differential equation,, subject to the initial condition and the boundary conditionsu. Let . It is established that if M>∞, then the set of quenching points is in for q>0, and in for q>0. At each quenching point, it is shown that ut bows up. The critical length is proved to be the same as that for q=0. A comparison of the quenching time with that for q=0 is also given.

On the global solutions to two-phase nonequilibrium problems

In this paper, we consider a Stefan-like problem with a kinetic condition on the free boundary, i.e. a two-phase nonequilibrium problem in one dimension, and prove that it has a unique classical and global solution under some conditions. We also investigate the travelling wave solutions.

The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation

The paper deals with the problem of solvability of the mixed problem for a linear second-order hyperbolic partial differential equation. The minimal necessary and sufficient conditions for the existence of a unique classical solution to this problem are established.

Radially symmetric global classical solutions of non‐linear wave equations

AbstractThe Cauchy problem for semilinear wave equations utt − Δu + h(|x|)up = 0 with radially symmetric smooth ‘large’ data has a unique global classical solution in arbitrary space dimensions if h is non‐negative and p any odd integer provided the smooth factor h vanishes with sufficiently high order at the origin and is bounded together with its derivatives.

On The Coupled Cahn-hilliard Equations

This paper is concerned with a system of nonlinear partial differential equations, in short, the coupled Cahn-Hilliard equations, which consists of a fourth order quasilinear parabolic equation and a second order quasilinear parabolic equation. This system was recently derived by Penrose and Fife and also by Alt and Pawlow to describe the non-isothermal phase separation of a two-component system. The global existence and uniqueness of classical solutions is proved. The results about the asymptotic behavior, as time goes to infinity, of solution and about the existence and multiplicity of solutions to the corresponding stationary problem, which is a nonlinear boundary value problem involving nonlocal term and constraints, are also obtained.

Generation and propagation of interfaces in reaction-diffusion systems

This paper is concerned with the asymptotic behavior, as ε ↘ 0 \varepsilon \searrow 0 , of the solution ( u ε , v ε ) ({u^\varepsilon },{v^\varepsilon }) of the second initial-boundary value problem of the reaction-diffusion system: \[ { u t ε − ε Δ u ε = 1 ε f ( u ε , υ ε ) ≡ 1 ε [ u ε ( 1 − u ε 2 ) − υ ε ] , υ t ε − Δ υ ε = u ε − γ υ ε \left \{ {\begin {array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \Delta {u^\varepsilon } = \frac {1}{\varepsilon }f({u^\varepsilon },{\upsilon ^\varepsilon }) \equiv \frac {1}{\varepsilon }[{u^\varepsilon }(1 - {u^{\varepsilon 2}}) - {\upsilon ^\varepsilon }],} \hfill \\ {\upsilon _t^\varepsilon - \Delta {\upsilon ^\varepsilon } = {u^\varepsilon } - \gamma {\upsilon ^\varepsilon }} \hfill \\ \end {array} } \right . \] where γ > 0 \gamma > 0 is a constant. When v ∈ ( − 2 3 / 9 , 2 3 / 9 ) v \in ( - 2\sqrt 3 /9,2\sqrt 3 /9) , f f is bistable in the sense that the ordinary differential equation u t = f ( u , v ) {u_t} = f(u,v) has two stable solutions u = h − ( v ) u = {h_ - }(v) and u = h + ( v ) u = {h_ + }(v) and one unstable solution u = h 0 ( v ) u = {h_0}(v) , where h − ( v ) , h 0 ( v ) {h_ - }(v), {h_0}(v) , and h + ( v ) {h_ + }(v) are the three solutions of the algebraic equation f ( u , v ) = 0 f(u,v) = 0 . We show that, when the initial data of v v is in the interval ( − 2 3 / 9 , 2 3 / 9 ) ( - 2\sqrt 3 /9,2\sqrt 3 /9) , the solution ( u ε , v ε ) ({u^\varepsilon },{v^\varepsilon }) of the system tends to a limit ( u , v ) (u,v) which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function u u is a "phase" function in the sense that it coincides with h + ( v ) {h_ + }(v) in one region Ω + {\Omega _ + } and with h − ( v ) {h_ - }(v) in another region Ω − {\Omega _ - } . The common boundary (free boundary or interface) of the two regions Ω − {\Omega _ - } and Ω + {\Omega _ + } moves with a normal velocity equal to V ( v ) \mathcal {V}(v) , where V ( ∙ ) \mathcal {V}( \bullet ) is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially u ( ∙ , 0 ) − h 0 ( v ( ∙ , 0 ) ) u( \bullet , 0) - {h_0}(v( \bullet , 0)) takes both positive and negative values, then an interface will develop in a short time O ( ε | ln ⁡ ε | ) O(\varepsilon |\ln \varepsilon |) near the hypersurface where u ( x , 0 ) − h 0 ( v ( x , 0 ) ) = 0 u(x,0) - {h_0}(v(x,0)) = 0 .

The two‐dimensional Wigner–Poisson problem for an electron gas in the charge neutral case

AbstractWe prove the global existence of a unique classical solution of the two‐dimensional Wigner–Poisson problem by a reformulation as a system of countably many Schrödinger equations coupled to a Poisson equation. The charge neutrality of the system implies the boundedness of the electrostatic potential in two dimensions, which is an important ingredient for the proof.

Local existence for solutions of fully non-linear wave equations

Let Ω be a domain in ℝn and let mϵ ℕ; be given. We study the initial-boundary value problem for the equation with a homogeneous Dirichlet boundary condition; here u is a scalar function, \documentclass{article}\pagestyle{empty}\begin{document}$ \bar D_x^m u: = (\partial _x^\alpha u)_{|\alpha | \le m} $\end{document} and certain restrictions are made on F guaranteeing that energy estimates are possible. We prove the existence of a value of T>0 such that a unique classical solution u exists on [0, T]×Ω. Furthermore, we show that T → ∞ if the data tend to zero.

Solution of the forced nonlinear schrödinger (nls) equation using pde techniques

It is shown that there exists a unique global classical solution u ∈ C⊃1⊃(L⊃2⊃) ∩ C°(H⊃2⊃) of the forced NLS equation (1.1) below for uo ∈ H2 and Q ∈ C2, with uo(0)=Q(0)and K real

Nonlinear filtering of reflecting diffusion processes

In the nonlinear filtering model y t=h t(X t)+e t, 0⩽t⩽T , where ( e 1 ) is a finitely additive white noise, the problem of finding the conditional density u( t, x) of X t given observations { y u :0⩽ u⩽ t} is considered when ( X t ) is a reflecting diffusion process. It is shown that u( t, x) can be obtained as the unique classical solution of an initial-boundary value problem for a parabolic PDE.