Abstract
Using heat kernel estimates obtained in [18] and the Feynman-Kac formula, we investigate finite-time blow-up and stability of semilinear partial differential equations of the form ∂w t /∂t (x) = Δw t (x) − V (x)w t (x) + v t (x)G(w t (x)), w 0(x) ≥ 0, x ∈ ℝd, where v and G are positive measurable functions subject to certain growth conditions, and V is a positive bounded potential. We recover the results of [19] and [14] by probabilistic arguments and in the quadratic decay case V (x) ∼+∞ a(1 + |x|2)−1, a > 0, we find two critical exponents β*(a), β*(a) with 0 < β*(a) ≤ β*(a) < 2/d, such that any nontrivial positive solution blows up in finite time if 0 < β < β*(a), whereas if β*(a) < β, then nontrivial positive global solutions may exist.
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© 2007 Birkhäuser Verlag Basel/Switzerland
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López-Mimbela, J.A., Privault, N. (2007). Critical Exponents for Semilinear PDEs with Bounded Potentials. In: Dalang, R.C., Russo, F., Dozzi, M. (eds) Seminar on Stochastic Analysis, Random Fields and Applications V. Progress in Probability, vol 59. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8458-6_14
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DOI: https://doi.org/10.1007/978-3-7643-8458-6_14
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