## Multidimensional viscoelasticity equations with nonlinear damping and source terms.(English)Zbl 1057.74007

The authors investigate the initial-boundary value problem $$u_{tt}-\Delta u_t-\sum_{i=1}^N\frac \partial{\partial x_i}\sigma_i(u_{x_i})+f(u_t)=g(u)$$, $$x\in \Omega$$, $$t>0,$$ $$u(x,0)=u_0(x)$$, $$u_t(x,0)=u_1(x)$$, $$x\in \Omega$$; $$u(x,t)=0$$, $$x\in \partial\Omega$$, $$t\geq 0,$$ where $$\Omega\subset \mathbb R^N$$ is a bounded domain. They introduce a potential well $$W=\{u\in W_0^{1,m+1}(\Omega)\mid I(u)>0$$, $$J(u)<d\}\cup \{0\}$$, where $$I(u)=B_1\| \nabla u\|_{m+1}^{m+1}- B_5\| u\|_{p+1}^{p+1}$$, $$J(u)=\frac {B_1}{m+1}\| \nabla u\|_{m+1}^{m+1}- \frac {B_5}{p+1}\| u\|_{p+1}^{p+1},$$ $$d=\inf J(u)$$ is subject to $$y\in W_0^{1,m+1}(\Omega)$$, $$\| \nabla u\|_{m+1}\neq 0$$, $$I(u)=0$$, and $$\sigma_i, f, g\in C$$ satisfy the conditions: $$\sigma_i(0)=0$$, $$(\sigma_i(s_1)-\sigma_i(s_2))(s_1-s_2)\geq 0$$ $$\forall s_1, s_2\in \mathbb R,$$ $$B_1| s|^m\leq | \sigma_i(s)| \leq B_2(1+| s|^m )$$, $$m\geq 1$$, $$i=1,\dots,N$$. $$f(s)s\geq 0$$, $$B_3 | s|^p\leq | f(s)| \leq B_4(1+| s|^q )$$, $$q\geq 1,$$ $$g(s)s\geq 0$$, $$| g(s)| \leq B_5| s|^p$$, $$1\leq m<p<\infty$$ if $$m+1\geq N$$; $$2\leq m+1<p+1\leq N(m+1)/(N-m-1)$$ if $$m+1<N.$$ Some new families of potential wells are introduced. Using them, the authors prove new existence theorems on global solutions, and examine the behaviour of vacuum isolation of solutions.

### MSC:

 74D10 Nonlinear constitutive equations for materials with memory 74H20 Existence of solutions of dynamical problems in solid mechanics 35L20 Initial-boundary value problems for second-order hyperbolic equations 35Q72 Other PDE from mechanics (MSC2000)

### Keywords:

potential well; existence; global solutions; vacuum isolation
Full Text:

### References:

 [1] Andrews, G., On the existence of solutions to the equation $$u_{ tt }$$−$$u_{ xxt }=σ(u_x)_{x$$ · Zbl 0397.35011 [2] Andrews, G.; Ball, J. M., Asymptotic behavior and changes in phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations, 44, 306-341 (1982) · Zbl 0501.35011 [3] Ang, D. D.; Dinh, P. N., Strong solutions of a quasilinear wave, equation with nonlinear damping, SIAM. J. Math. Anal., 102, 337-347 (1988) · Zbl 0662.35072 [4] Clements, J., Existence theorems for a quasilinear evolution equation, SIAM. J. Appl. Math., 226, 745-752 (1974) · Zbl 0252.35044 [5] Clements, J., On the existence and uniqueness of solutions of the equation $$u_{ tt }$$−$$(∂/∂x_i)σ_{i$$ · Zbl 0312.35017 [6] Dafermos, C. M., The mixed initial-boundary value problem for the equations of nonlinear one-dimensional visco-elasticity, J. Differential Equations, 6, 71-86 (1969) · Zbl 0218.73054 [7] Davis, P., A quasi-linear hyperbolic and related third-order equation, J. Math. Anal. Appl., 51, 596-606 (1975) · Zbl 0312.35018 [8] Engler, H., Strong solutions for strongly damped quasilinear wave equations, Contemp. Math., 64, 219-237 (1987) · Zbl 0638.35054 [9] Greenberg, J. M., On the existence, uniqueness and stability of solutions of the equation $$ρ_0X_{ xx }=E(X_x)X_{ xx }+λX_{ xxt }$$, J. Math. Anal. Appl., 25, 575-591 (1969) · Zbl 0192.44803 [10] Greenberg, J. M.; MacCamy, R. C., On the exponential stability of solutions of $$E(u_x)u_{ xx }+λu_{ xtx }=ρu_{ tt }$$, J. Math. Anal. Appl., 31, 406-417 (1970) · Zbl 0219.35010 [11] Greenberg, J. M.; MacCamy, R. C.; Mizel, J. J., On the existence, uniqueness and stability of the solutions of the equation $$σ(u_x)u_{ xx } +λu_{ xxt }=ρ_0u_{ tt }$$, J. Math. Mech., 17, 707-728 (1968) · Zbl 0157.41003 [12] Lions, J. L., Quelques methods de resolution des problems aux limits nonlinears (1969), Dunod: Dunod Paris · Zbl 0189.40603 [13] Yacheng, L., On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192, 155-169 (2003) · Zbl 1024.35078 [14] Yacheng, L.; Dacheng, L., Initial boundary value problem, periodic boundary problem and initial value problem of equation $$u_{ tt }=u_{ xxt }+σ(u_x)_{x$$ · Zbl 0703.35118 [15] Yamada, Y., Some remarks on the equation $$Y_{ tt }$$−$$ρ(y_x)y_{ xx }$$−$$y_{ xtx }=f$$, Osaka. J. Math., 17, 303-323 (1980) · Zbl 0446.35071 [16] Zhijian, Y., Initial-boundary value problem and Cauchy problem for a quasilinear evolution equation, Acta Math. Sci., 19, Supp, 52-61 (1999) [17] Zhijian, Y., Existence and asymptotic behaviour of solutions for a class of quasilinear evolution equations with nonlinear damping and source terms, Math. Meth. Appl. Sci., 25, 795-814 (2002) · Zbl 1011.35121 [18] Zhijian, Y.; Changming, S., Blow-up of solutions for a class of quasilinear evolution equations, Nonlinear Anal. TMA, 8, 2017-2032 (1977) · Zbl 0872.35015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.