Absence of global solutions to a class of nonlinear parabolic inequalities

M. Guedda. "Absence of global solutions to a class of nonlinear parabolic inequalities." Colloquium Mathematicae 94.2 (2002): 195-220. <http://eudml.org/doc/284556>.

@article{M2002,
abstract = {We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_\{t\} ≥ -(-Δ)^\{β/2\}u - V(x)u + h(x,t)u^\{p\}$, where $(-Δ)^\{β/2\}$, 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying $V₊(x) ∼ a|x|^\{-b\}$, where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0. We show that the existence of solutions depends on the behavior at infinity of both initial data and h. In addition to our main results, we also discuss the nonexistence of solutions for some degenerate parabolic inequalities like $u_\{t\} ≥ Δu^\{m\} + u^\{p\}$ and $u_\{t\} ≥ Δ_\{p\}u + h(x,t)u^\{p\}$. The approach is based upon a duality argument combined with an appropriate choice of a test function. First we obtain an a priori estimate and then we use a scaling argument to prove our nonexistence results.},
author = {M. Guedda},
journal = {Colloquium Mathematicae},
keywords = {absence of nonnegative global solutions; parabolic inequalities; degenerate parabolic inequalities; nonexistence},
language = {eng},
number = {2},
pages = {195-220},
title = {Absence of global solutions to a class of nonlinear parabolic inequalities},
url = {http://eudml.org/doc/284556},
volume = {94},
year = {2002},
}

TY - JOUR
AU - M. Guedda
TI - Absence of global solutions to a class of nonlinear parabolic inequalities
JO - Colloquium Mathematicae
PY - 2002
VL - 94
IS - 2
SP - 195
EP - 220
AB - We study the absence of nonnegative global solutions to parabolic inequalities of the type $u_{t} ≥ -(-Δ)^{β/2}u - V(x)u + h(x,t)u^{p}$, where $(-Δ)^{β/2}$, 0 < β ≤ 2, is the β/2 fractional power of the Laplacian. We give a sufficient condition which implies that the only global solution is trivial if p > 1 is small. Among other properties, we derive a necessary condition for the existence of local and global nonnegative solutions to the above problem for the function V satisfying $V₊(x) ∼ a|x|^{-b}$, where a ≥ 0, b > 0, p > 1 and V₊(x): = maxV(x),0. We show that the existence of solutions depends on the behavior at infinity of both initial data and h. In addition to our main results, we also discuss the nonexistence of solutions for some degenerate parabolic inequalities like $u_{t} ≥ Δu^{m} + u^{p}$ and $u_{t} ≥ Δ_{p}u + h(x,t)u^{p}$. The approach is based upon a duality argument combined with an appropriate choice of a test function. First we obtain an a priori estimate and then we use a scaling argument to prove our nonexistence results.
LA - eng
KW - absence of nonnegative global solutions; parabolic inequalities; degenerate parabolic inequalities; nonexistence
UR - http://eudml.org/doc/284556
ER -