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:
$a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1})\ \ \ \forall t<T,\ c_0=\textrm{const}>0a_1(t)\geqslant c_0\exp(-\omega(T-t)(T-t)^{-1})\ \ \ \forall t<T,\ c_0=\textrm{const}>0$
with some continuous nondecreasing function
$\omega(\tau)\geqslant0$
$\forall\tau>0$
. Under additional condition:
$\omega(\tau)\rightarrow\omega_0=\textrm{const}>0\ \ \ \text{ as }\tau\rightarrow0$
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 & 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. Google Scholar

[2]

Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies, Trans. Amer. Math. Soc., 364 (2012), 6039-6070. Google Scholar

[3]

Y. DuR. 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. Google Scholar

[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. Google Scholar

[5]

A. A. KovalevskyI. 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. Google Scholar

[6]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967,736 p.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. Google Scholar

[2]

Y. Du and R. Peng, The periodic logistic equation with spatial and temporal degeneracies, Trans. Amer. Math. Soc., 364 (2012), 6039-6070. Google Scholar

[3]

Y. DuR. 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. Google Scholar

[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. Google Scholar

[5]

A. A. KovalevskyI. 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. Google Scholar

[6]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967,736 p.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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