This paper investigates the asymptotic decay behavior of solutions to Caputo-type fractional evolution equations with two-parameter damping of the form $ ^{C}\!D_t^{\gamma}u(t)+k\, ^{C}\!D_t^{\delta}u(t) = Au(t) $, where $ 0 < \delta < \gamma\leq 1 $ and $ k > 0 $. The characteristic equation $ \lambda^\gamma+k\lambda^\delta = \mu $ ($ \mu < 0 $) has no roots in the cut plane $ \mathbb{C}\setminus(-\infty, 0] $, so the resolvent family is represented entirely by a branch cut integral. Exploiting the spectral representation of the Caputo fractional $ (\gamma, \delta, k) $ resolvent family and Watson's lemma, we establish $ \|U(t)\|\leq C(1+t)^{-\delta} $ for self-adjoint $ A $ with compact resolvent. For sectorial generators with the compatibility condition $ \omega < \pi(1-\gamma) $, the same decay rate $ \|U(t)A^{-1}\|\leq Ct^{-\delta} $ holds. Exponential decay occurs only in the excluded case $ \gamma = \delta = 1 $; the lower-order fractional derivative $ \delta $ governs the long-time polynomial decay.
Citation: Shi-you Lin, Ting-ting Hu, Zhi-chao Lu. Decay rates for Caputo-type fractional evolution equations with two-parameter damping[J]. AIMS Mathematics, 2026, 11(9): 31664-31709. doi: 10.3934/math.20261248
This paper investigates the asymptotic decay behavior of solutions to Caputo-type fractional evolution equations with two-parameter damping of the form $ ^{C}\!D_t^{\gamma}u(t)+k\, ^{C}\!D_t^{\delta}u(t) = Au(t) $, where $ 0 < \delta < \gamma\leq 1 $ and $ k > 0 $. The characteristic equation $ \lambda^\gamma+k\lambda^\delta = \mu $ ($ \mu < 0 $) has no roots in the cut plane $ \mathbb{C}\setminus(-\infty, 0] $, so the resolvent family is represented entirely by a branch cut integral. Exploiting the spectral representation of the Caputo fractional $ (\gamma, \delta, k) $ resolvent family and Watson's lemma, we establish $ \|U(t)\|\leq C(1+t)^{-\delta} $ for self-adjoint $ A $ with compact resolvent. For sectorial generators with the compatibility condition $ \omega < \pi(1-\gamma) $, the same decay rate $ \|U(t)A^{-1}\|\leq Ct^{-\delta} $ holds. Exponential decay occurs only in the excluded case $ \gamma = \delta = 1 $; the lower-order fractional derivative $ \delta $ governs the long-time polynomial decay.
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Shi-you Lin, Ting-ting Hu, Zhi-chao Lu. Decay rates for Caputo-type fractional evolution equations with two-parameter damping[J]. AIMS Mathematics, 2026, 11(9): 31664-31709. doi: 10.3934/math.20261248
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