Initial-boundary value problems for the equation $u_{tt}=(\sigma (u_x))_x+(\alpha (u_x)u_{xt})_x+f$
Authors:
K. Kuttler and D. Hicks
Journal:
Quart. Appl. Math. 46 (1988), 393-407
MSC:
Primary 35L70; Secondary 35Q20, 73F15
DOI:
https://doi.org/10.1090/qam/963578
MathSciNet review:
963578
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Abstract: Existence and uniqueness theorems are proved for global weak solutions of initial-boundary value problems corresponding to the equation \[ {u_{tt}} = {\left ( {\sigma \left ( {{u_x}} \right )} \right )_x} + {\left ( {\alpha \left ( {{u_x}} \right ){u_{xt}}} \right )_x} + f\] under assumptions that do not require smoothness or monotonicity of $\sigma$. The initial data are not assumed to be smooth, the boundary data are allowed to be time dependent, and $f$ is only assumed to be in ${L^2}$.
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E. C. Aifantis and J. B. Serrin, The mechanical theory of fluid interfaces and Maxwell’s rule, J. Colloid Interface Sci. 96, 517–529 (1983)
G. Andrews, On the existence of solutions to the equation ${u_{tt}} = {u_{xxt}} + \sigma {\left ( {{u_x}} \right )_x}$, J. Differential Equations 35, 200–231 (1980)
G. Andrews and J. M. Ball, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations 44, 306–341 (1982)
C. Dafermos, The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity, J. Differential Equations 6, 71–86 (1969)
J. M. Greenberg, On the existence, uniqueness, and stability of solutions of the equation ${\rho _0}{X_{tt}} = \\ E\left ( {{X_x}} \right ){X_{xx}} + \lambda {X_{xxt}}$, J. Math. Anal. Appl. 25, 575–591 (1969)
J. M. Greenberg and R. C. MacCamy, On the exponential stability of solutions of $E\left ( {{u_x}} \right ){u_{xx}} + \\ \lambda {u_{xtx}} = \rho {u_{tt}}$, J. Math. Anal. Appl. 31, 406–417 (1970)
J. M. Greenberg, R. C. MacCamy, and V. J. Mizel, On the existence, uniqueness, and stability of solutions of the equation $\sigma ’\left ( {{u_x}} \right ){u_{xx}} + \lambda {u_{xtx}} = {\rho _0}{u_{tt}}$, J. Math. Mech. 17, 707–728 (1968)
Ya Kanel, Certain systems of quasi-linear parabolic equations of divergence type, USSR Comput. Math. and Math. Phys. 6, 74–88 (1966)
Ya Kanel, A model system of equations for the one-dimensional motion of a gas, Differential Equations 4, 374–380 (1968) (English translation)
A. V. Kazhikhov, Correctness “in the large” of mixed-boundary value problems for a model system of equations of a viscous gas, Dinamika Splošn. Sredy 21, 18–47 (1975) (in Russian)
A. V. Kazhikhov and V. B. Nikolaev, Correctness of boundary value problems for equations of gas dynamics with a nonmonotone state function, Čisl. Metody Meh. Splošn. Sredy 10, 77–84 (1979) (in Russian)
K. Kuttler and D. Hicks, Initial-boundary value problems for some nonlinear conservation laws, Applicable Anal. 24, 1–12 (1987)
K. Kuttler, Regularity of weak solutions of some nonlinear conservation laws, Applicable Analysis, Vol. 26, October 1987
K. Kuttler, Time dependent implicit evolution equations, Nonlinear Anal. Theory, Methods Appl. 10(5), 447–463 (1986)
K. Kuttler, Initial boundary value problems for the displacement in an isothermal viscous gas, submitted
K. Kuttler and E. Aifantis, Existence and uniqueness in nonclassical diffusion, Quart. Appl. Math. 45, 549–560 (1987)
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969
R. Pego, Phase transitions in one-dimensional nonlinear visco-elasticity: admissibility and stability, Arch. Rat. Mech. Anal. V. 97(4), 353–394 (1987)
R. E. Showalter, Hilbert space methods for partial differential equations, Pittman, 1977
V. A. Solonnikov and A. V. Kazhikhov, Existence theorems for the equations of motion of a compressible viscous fluid, Annual Rev. Fluid Mech. 13, 79–95 (1981)
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 1977
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Article copyright:
© Copyright 1988
American Mathematical Society