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Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations. (English) Zbl 0595.35058
The following problems are studied: \[ (E_ 1)\quad u_ t- \sum^{N}_{i=1}(| u_{x_ i}|^ m u_{x_ i})_{x_ i}+\beta (x,u)=0,\quad u(x,0)=u_ 0\in W_ 0^{1,m+2} \]
\[ and\quad E_ 2\quad u_ t-\Delta (| u|^ m u)+\beta (x,u)=0,\quad u(x,0)=u_ 0\in W_ 0^{1,m+2} \] in a bounded domain for the x- variable in \(R^ N\), with locally Hölder continuous \(\beta\) and \(| \beta (x,u)| \leq k_ 0 | u|^{\alpha +1}.\)
The author proves the global existence and a priori estimates of the solutions provided the norm of initial data is sufficiently small in \(L^{p+2}\) with p depending on N, m and \(\alpha\).
Reviewer: B.Nowak

35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B45 A priori estimates in context of PDEs
35B20 Perturbations in context of PDEs
Full Text: DOI
[1] Alikakos, N.D., Lp-bounds of solutions of reaction diffusion equations, Communs partial diff. eqns., 4, 868-927, (1979) · Zbl 0421.35009
[2] Brezis, H.; Crandall, M.G., Uniqueness of solutions of the initial value problems for ut - δφ(u) = 0, J. math. pures appl., 58, 153-163, (1979) · Zbl 0408.35054
[3] Fujita, H., On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Proc. symp. pure math., Nonlinear functional analysis, ann. am. math. soc., 18, (1968)
[4] Galaktinov, V.A., A boundary value problem for the nonlinear parabolic equation ut = δuα-1+up, Diff. eqns, 17, 551-555, (1981), (In Russian.)
[5] Ishi, H., Asymptotic stability and blowing up of solutions of some nonlinear evolution equations, J. diff. eqns., 26, 291-319, (1977)
[6] Ladyzhenskaja, O.A.; Solonnikov, V.A.; Ural’tseva, N.N., Linear and quasi-linear equations of parabolic type, (1968), Am. Math. Soc Providence, R.I · Zbl 0174.15403
[7] Levine, H.A.; Sacks, P.E., Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. diff. eqns., 33, 135-161, (1984) · Zbl 0487.34003
[8] Nakao, M., On solutions of perturbed porous medium equations, (), 539-547
[9] Nakao, M., On solutions to the initial-boundary value problems for ∂/∂Tu − δβ(u) = f, J. math. soc. Japan, 35, 71-83, (1983) · Zbl 0494.35055
[10] Nakao, M., Lp-estimates of solutions of some nonlinear degenerate diffution equations, J. math. soc. Japan, 37, 41-63, (1985)
[11] Nakao M., Global existence and smoothing effect for a semilinear parabolic equation with a nonmonotonic perturbation, Kunkcialaj Ekvacioj (to appear). · Zbl 0619.35056
[12] Nakao, M.; Narazaki, T., Existence and decay of solutions of some nonlinear parabolic variational inequalities, Int. J. math. & math. sci., 2, 79-102, (1980) · Zbl 0451.35036
[13] Ôtani, M., On existence of strong solutions for du/dt+ϑψ1(u)− ϑψ2(u) ∋f, J. fac. sci. univ. Tokyo, sec. IA. math., 24, 575-605, (1977) · Zbl 0386.47040
[14] Ôtani, M., Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, J. diff. eqns., 46, 268-299, (1982), Cauchy Problems · Zbl 0495.35042
[15] Tsutsumi, M., Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. R.I.M.S. Kyoto univ., 8, 211-229, (1972-1973) · Zbl 0248.35074
[16] Sacks, P.E., Global behavior for a class of nonlinear evolution equations, SIAM J. math. analysis, 16, 233-249, (1985) · Zbl 0572.35062
[17] Weissler, F.B., Semilinear evolution equations in Banach spaces, J. funct. analysis, 32, 277-296, (1979) · Zbl 0419.47031
[18] Weissler, F.B., Existence and non-existence of global solutions for a semilinear heat equation, Israel J. math., 38, 29-40, (1981) · Zbl 0476.35043
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