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1995
In Theorem 2.1.10 we saw that if A is the infinitesimal generator of a C0-semigroup T(t), the solution of the abstract homogeneous Cauchy initial value problem $$\begin{array}{*{20}{c}}{\dot z\left( t \right) = Az\left( t \right),}&{t \geqslant 0,}&{z\left( 0 \right) = {z_0} \in D\left( A \right)}\end{array}$$ is given by $$z\left( t \right)
Ruth F. Curtain, Hans Zwart
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In Theorem 2.1.10 we saw that if A is the infinitesimal generator of a C0-semigroup T(t), the solution of the abstract homogeneous Cauchy initial value problem $$\begin{array}{*{20}{c}}{\dot z\left( t \right) = Az\left( t \right),}&{t \geqslant 0,}&{z\left( 0 \right) = {z_0} \in D\left( A \right)}\end{array}$$ is given by $$z\left( t \right)
Ruth F. Curtain, Hans Zwart
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Applied Mathematics and Computation, 2004
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On the Nonlinear Cauchy Problem
2006Our aim is to describe how to obtain, with the same procedure, several results of local existence, uniqueness and propagation of regularity for the solution of a quasilinear hyperbolic Cauchy Problem.
CICOGNANI, Massimo, ZANGHIRATI, Luisa
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2009
The theory of the Cauchy problem for hyperbolic conservation laws is confronted with two major challenges. First, classical solutions, starting out from smooth initial values, spontaneously develop discontinuities; hence, in general, only weak solutions may exist in the large. Next, weak solutions to the Cauchy problem fail to be unique.
Li Tatsien, Wang Libin
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The theory of the Cauchy problem for hyperbolic conservation laws is confronted with two major challenges. First, classical solutions, starting out from smooth initial values, spontaneously develop discontinuities; hence, in general, only weak solutions may exist in the large. Next, weak solutions to the Cauchy problem fail to be unique.
Li Tatsien, Wang Libin
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1997
Suppose L 1 and L 2 are topological vector spaces and T : L 1 → L 2 is a continuous linear mapping. We consider the aspect of Problem 7.1.1 connected with describing the range of the mapping T.
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Suppose L 1 and L 2 are topological vector spaces and T : L 1 → L 2 is a continuous linear mapping. We consider the aspect of Problem 7.1.1 connected with describing the range of the mapping T.
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On the Cauchy and Cauchy–Darboux problems for semilinear wave equations
Georgian Mathematical Journal, 2014Abstract The Cauchy and Cauchy–Darboux problems for semilinear wave equations in the class of continuous functions are investigated. The questions of existence, uniqueness and nonexistence of global solutions of the problems are considered. The local solvability of the problems is also discussed.
Jokhadze, Otar, Kharibegashvili, Sergo
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The Cauchy Problem for the Heat Equation
SIAM Journal on Numerical Analysis, 1967Part I is devoted to the consideration of a Cauchy-like problem for the heat equation. Let $u(x,t)$ satisfy the heat equation in $D = \{ (x,t):0 < x < s(t),0 < t \leqq T\} $ and let $u(x,0) = \varp...
Cannon, J. R., Douglas, Jim jun.
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On the cauchy problem for the einstein equation
Doklady Mathematics, 2012zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Lychagin, V. V., Yumaguzhin, V. A.
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Fractional Abstract Cauchy Problems
Integral Equations and Operator Theory, 2011Fractional abstract Cauchy problems with order \(\alpha \in (1,2)\) are studied. Existence and uniqueness of mild solutions and strong solutions of the inhomogeneous fractional Cauchy problem are established.
Li, Kexue, Peng, Jigen
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On Cauchy’s problem in general relativity
Il Nuovo Cimento, 1959It is shown that a suitable set of Cauchy’s conditions for Einstein’s equations in vacuo consists in specifying the values ofgkl andgkl,0 on a hypersurfacex0 = 0. The correspondingg0k are to be found by solving threespatial second order differential equations, andg00 is then given by analgebraic relation.
Peres, A., Rosen, N.
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