# American Institute of Mathematical Sciences

August  2017, 10(4): 895-907. doi: 10.3934/dcdss.2017045

## Large solutions of parabolic logistic equation with spatial and temporal degeneracies

 1 Institute of Applied Mathematics and Mechanics, The National Academy of Sciences of Ukraine, Dobrovol'skogo str. 1, Slavyansk, Donetsk region, 84116, Ukraine 2 Peoples' Friendship University of Russia, Miklukho-Maklaya str. 6, Moscow, 117198, Russia

Received  May 2016 Revised  October 2016 Published  April 2017

There is studied asymptotic behavior as
 $t\rightarrow T$
of arbitrary solution of equation
 $P_0(u):=u_t-\Delta u=a(t,x)u-b(t,x)|u|^{p-1}u\ \ \ \text{ in } [0,T)\times\Omega,$
where
 $\Omega$
is smooth bounded domain in
 $\mathbb{R}^N$
,
 $0 < T < \infty$
,
 $p>1$
,
 $a(\cdot)$
is continuous,
 $b(\cdot)$
is continuous nonnegative function, satisfying condition:
 $b(t, x)\geqslant a_1(t)g_1(d(x))$
,
 $d(x):=\textrm{dist}(x, \partial\Omega)$
. Here
 $g_1(s)$
is arbitrary nondecreasing positive for all
 $s>0$
function and
 $a_1(t)$
satisfies:
 $a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1})\ \ \ \forall t0a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1})\ \ \ \forall t0$
with some continuous nondecreasing function
 $\omega(\tau)\geqslant0$
 $\forall\tau>0$
 $\omega(\tau)\rightarrow\omega_0=\textrm{const}>0\ \ \ \text{ as }\tau\rightarrow0$
it is proved that there exist constant
 $k:0 < k < \infty$
, such that all solutions of mentioned equation (particularly, solutions, satisfying initial-boundary condition
 $u|_\Gamma=\infty$
, where
 $\Gamma=(0, T)\times\partial\Omega\cup\{0\}\times\Omega$
) stay uniformly bounded in
 $\Omega_0:=\{x\in\Omega:d(x)>k\omega_0^{\frac12}\}$
as
 $t\rightarrow T$
. Method of investigation is based on local energy estimates and is applicable for wide class of equations. So in the paper there are obtained similar sufficient conditions of localization of singularity set of solutions near to the boundary of domain for equation with main part
 $P_0(u)=(|u|^{\lambda-1}u)_t-\sum_{i=1}^N(|\nabla_xu|^{q-1}u_{x_i})_{x_i}$
if
 $0 < \lambda\leqslant q < p$
.
Citation: Andrey Shishkov. Large solutions of parabolic logistic equation with spatial and temporal degeneracies. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 895-907. doi: 10.3934/dcdss.2017045
##### References:
 [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. [2] Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies, Trans. Amer. Math. Soc., 364 (2012), 6039-6070. [3] Y. Du, R. Peng and P. Polachik, The parabolic logistic equation with blow-up initial and boundary values, Journal D'Analyse Mathematique, 118 (2012), 297-316.  doi: 10.1007/s11854-012-0036-0. [4] J. L. Diaz and L. Veron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc., 290 (1985), 787-814. [5] A. A. Kovalevsky, I. I. Skrypnik and A. E. Shishkov, Singular Solutions in Nonlinear Elliptic and Parabolic Equations. De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Basel, 24 (2016), 435 p. [6] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967,736 p. [7] V. A. Galaktionov and A. E. Shishkov, Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh. Sect. A,, 133 (2003), 1075-1119. [8] V. A. Galaktionov and A. E. Shishkov, Self-similar boundary blow-up for higher-order quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh. Sect. A,, 135 (2005), 1195-1227. [9] A. E. Shishkov and A. G. Shchelkov, Boundary regimes with peaking for general quasilinear parabolic equations in multidimensional domains, Sb. Math., 190 (1999), 447-479.

show all references

##### References:
 [1] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. [2] Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies, Trans. Amer. Math. Soc., 364 (2012), 6039-6070. [3] Y. Du, R. Peng and P. Polachik, The parabolic logistic equation with blow-up initial and boundary values, Journal D'Analyse Mathematique, 118 (2012), 297-316.  doi: 10.1007/s11854-012-0036-0. [4] J. L. Diaz and L. Veron, Local vanishing properties of solutions of elliptic and parabolic quasilinear equations, Trans. Amer. Math. Soc., 290 (1985), 787-814. [5] A. A. Kovalevsky, I. I. Skrypnik and A. E. Shishkov, Singular Solutions in Nonlinear Elliptic and Parabolic Equations. De Gruyter Series in Nonlinear Analysis and Applications, De Gruyter, Basel, 24 (2016), 435 p. [6] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967,736 p. [7] V. A. Galaktionov and A. E. Shishkov, Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh. Sect. A,, 133 (2003), 1075-1119. [8] V. A. Galaktionov and A. E. Shishkov, Self-similar boundary blow-up for higher-order quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh. Sect. A,, 135 (2005), 1195-1227. [9] A. E. Shishkov and A. G. Shchelkov, Boundary regimes with peaking for general quasilinear parabolic equations in multidimensional domains, Sb. Math., 190 (1999), 447-479.
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