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Absence of global solutions to a class of nonlinear parabolic inequalities

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Guedda M.

Colloquium Mathematicum

2002

tom 94

nr 2

195-220

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): = max{V(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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Guedda M.

1 2002

Wyniki wyszukiwania

Absence of global solutions to a class of nonlinear parabolic inequalities