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Article Contents

# On some quasilinear parabolic equations with non-monotone multivalued terms

• *Corresponding author: Vasile Staicu

Dedicated to Professor Viorel Barbu on the occasion of his 80th birthday

The first author is partially supported by the Grant-in-Aid for Scientific Research, #18K03382, the Ministry of Education, Culture, Sports, Science and Technology, Japan.
The second author is partially supported by Portuguese funds through CIDMA and the Portuguese Foundation for Science and Technology (FCT), within the project UID/MAT/04106/2013

• In this paper, we study the existence of solutions to the initial-boundary value problem for the following parabolic differential inclusion:

$\begin{equation*} \begin{cases} u_t\left(t, x \right) -\triangle _{p}u\left( t, x \right) \in -\partial \phi \left( u\left( t, x \right) \right) + G\left( t, x, u\left( t, x \right) \right) & (t, x) \in Q_T, \\ u(t, x) = 0 & (t, x) \in \Gamma_T, \\ u(0, x) = u_0(x) & x \in \Omega, \end{cases} \end{equation*}$

where $\Omega$ is a bounded open subset of $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega,$ $T>0$, $Q_{T}: = [0, T] \times \Omega$, $\Gamma_T: = [0, T] \times \partial\Omega$, $u_t = \frac{\partial u}{\partial t}$, $\triangle_{p}$ is the $p$-Laplace differential operator, $\partial \phi$ denotes the subdifferential of a proper lower semicontinuous convex function $\phi :\mathbb{R}\rightarrow \left[ 0, \infty \right]$, and $G:Q_{T}\times \mathbb{R}\rightarrow 2^{ \mathbb{R}} \backslash \{\emptyset\}$ is a nonmonotone multivalued mapping.

The case where $\phi \equiv 0$ and $G(t, x, u) = |u|^{q-2}u$ gives the prototype of our problem, denoted by (E)$_p$. The existence of time-local strong solutions for (E)$_p$ is already studied by several authors. However, these results require a stronger assumption on $q$ than that for the semi-linear case (E)$_p$ with $p = 2$.

More precisely, it has been long conjectured that (E)$_p$ should admit a time-local strong solution for the Sobolev-subcritical range of $q$, i.e., for all $q \in (2, p^\ast)$ with $p^\ast = \infty$ for $p \geq N$ and $p^\ast = \frac{N p}{N-p}$ for $p<N$, which is the well-known fact for the semi-linear case (E)$_p$ with $p = 2$.

The main purpose of the present paper is to show this conjecture holds true and to extend this classical study to the cases where $u \mapsto G(\cdot, \cdot, u)$ is upper semicontinuous or lower semicontinuous, each one is a generalized notion of the continuity in the theory of multivalued analysis.

We also discuss the extension of large or small local solutions along the lines of arguments developed in [28].

Mathematics Subject Classification: Primary: 35K20, 35A16, 35K92, 35R70, 35D35.

 Citation:

•  [1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, 1984. doi: 10.1007/978-3-642-69512-4. [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publ., Leyden, 1976. [3] G. Barletta, Parabolic equations with discontinuous nonlinearities, Bull. Australian Math. Soc., 63 (2001), 219-228.  doi: 10.1017/S0004972700019286. [4] H. Brézis, Opérateurs Maximaux Monotones et Semigroupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam, 1973. [5] H. Brézis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983. [6] H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach space, Commun. Pure Appl. Math., 23 (1970), 123-144.  doi: 10.1002/cpa.3160230107. [7] S. Carl, C. Grossmann and C. V. Pao, Existence and monotone iterations for parabolic differential inclusions, Commun. Appl. Nonlinear Anal., 3 (1996), 1-24. [8] S. Carl, Enclosure of solution for quasilinear dynamic hemivariational inequalities, Nonlinear World, 3 (1996), 281-298. [9] S. Carl and S. Heikkila, On a parabolic boundary value problem with discontinuous nonlinearity, Nonlinear Anal., 15 (1990), 1091-1095.  doi: 10.1016/0362-546X(90)90156-B. [10] S. Carl, V. K. Le and D. Motreanu, Nonlinear Variational Problems and Their Inequalities, Springer, New York, 2007. doi: 10.1007/978-0-387-46252-3. [11] S. Carl and D. Motreanu, Extremality in solving general quasilinear parabolic inclusions, J. Optim. Th. Appl., 123 (2004), 463-477.  doi: 10.1007/s10957-004-5718-z. [12] S. Carl and D. Motreanu, Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient, J. Differ. Equ., 191 (2003), 206-233.  doi: 10.1016/S0022-0396(03)00022-6. [13] T. Cardinali and N. S. Papageorgiou, Periodic problems and problems with discontinuities for nonlinear parabolic equations, Czech. Math. J., 50 (2000), 467-497.  doi: 10.1023/A:1022873208183. [14] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.  doi: 10.1016/0022-247X(81)90095-0. [15] H. Deguchi, On weak solutions of parabolic initial value problems with discontinuous nonlinearities, Nonlinear Anal., 63 (2005), 1-9.  doi: 10.1016/j.jmaa.2005.10.052. [16] N. Dunford and J. T. Schwartz, Linear Operators, Part. I, Interscience, New York, 1958. [17] E. Feireisl, A note on uniqueness for parabolic problems with discontinuous nonlinearities, Nonlinear Anal., 19 (1991), 1053-1056.  doi: 10.1016/0362-546X(91)90106-B. [18] E. Feireisl and J. Norbury, Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities, Proc. Royal Soc. Edinb., 119 (1991), 1-17.  doi: 10.1017/S0308210500028262. [19] B. A. Fleishman and T. J. Mahar, A step function model in chemical reactor theory: Multiplicity and stability of solutions, Nonlinear Anal., 15 (1981), 645-654.  doi: 10.1016/0362-546X(81)90080-8. [20] A. Fryszkowski, Continuous selections for a class of nonconvex multivalued maps, Stud. Math., 76 (1981), 163-174.  doi: 10.4064/sm-76-2-163-174. [21] L. Gasinski and  N. S. Papageorgiou,  Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman Hall and CRC Press, Boca Raton, 2005. [22] N. Halidias and N. S. Papageorgiou, Existence of solutions for nonlinear parabolic problems, Arch. Math. (Brno), 35 (1999), 255-274. [23] C. J. Himmelberg, Measurable relations, Fund. Math., 87 (1975), 53-72.  doi: 10.4064/fm-87-1-53-72. [24] J. Hofbauer and P. L. Simon, An existence theorem for parabolic equation on $\mathbb{R}^{N}$ with discontinuous nonlinearities, Electron. J. Qual. Theor. Differ. Equ., 15 (1981), 645-654. [25] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4. [26] H. Ishii, Asymptotic stability and blowing up of solutions of some nonlinear equations, J. Differ. Equ., 26 (1977), 291-319.  doi: 10.1016/0022-0396(77)90196-6. [27] M. Ôtani, On existence of strong solutions for $\frac{d u}{d t}(t) + \partial\psi^{1}(u(t)) - \partial\psi^{2}(u(t)) \ni f(t)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 575-605. [28] M. Ôtani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differ. Equ., 46 (1982), 268-299.  doi: 10.1016/0022-0396(82)90119-X. [29] M. Ôtani, Non-monotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Periodic problems, J. Differ. Equ., 54 (1984), 248-273.  doi: 10.1016/0022-0396(84)90161-X. [30] M. Ôtani, $L^{\infty}$-energy method, basic tools and usage, Differential Equations, Chaos and Variational Problems, Progress in Nonlinear Differential Equations and Their Applications, Vol. 75, Ed. by Vasile Staicu, Birkhauser, (2007), 357-376. doi: 10.1007/978-3-7643-8482-1_27. [31] M. Ôtani and V. Staicu, Existence results for quasilinear elliptic equations with multivalued nonlinear terms, Set-Valued Var. Anal., 22 (2014), 859-877.  doi: 10.1007/s11228-014-0289-0. [32] M. Ôtani and V. Staicu, On some nonlinear parabolic equations with nonmonotone multivalued terms, J. Convex Anal., 28 (2021), 771-794. [33] N. S. Papageorgiou, On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities, J. Math. Anal. Appl., 205 (1997), 434-453.  doi: 10.1006/jmaa.1997.5208. [34] J. Rauch, Discontinuous semilinear differential equations and multiple valued maps, Proc. Amer. Math. Soc., 64 (1977), 277-282.  doi: 10.2307/2041442. [35] J. Simon, Compact sets in the space $L^{p}(0, t; B)$, Ann. Mat. Pur. Appl., 146 (1986), 65-96.  doi: 10.1007/BF01762360. [36] M. Tsutsumi, Existence and nonexistence of global solutions for nonlinear parabolic equations, RIMS, Kyoto Univ., 17 (1972), 211-229.  doi: 10.2977/prims/1195193108.