A new critical exponent for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911623447314432 |
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| author | Phan, Duc An |
| author_facet | Phan, Duc An |
| contents | In this paper, we investigate the critical exponent for a semi-linear damped wave equation involving a Hartree-type nonlinearity of the form $\mathcal{I}_γ\left(|u|^{p_1}\right)|u|^{p_2}, p_1, p_2>0, γ\in[0, n)$, with initial data taken in the homogeneous Besov spaces $\dot{B}_{2, \infty}^{-β}$, where $β\in\left[0, \frac{n}{2}\right)$. Our approach is based on deriving decay estimates for solutions to the associated linear damped wave equation with initial data belonging to $\dot{B}_{2, \infty}^{-β}$, combined with refined tools from Harmonic Analysis. As a consequence, we identify a new critical exponent given by $$ p_1+p_2:=p_{\mathrm{Fuji}}\left(\tfrac{n+2β}{2+γ}\right):=1+\tfrac{4+2γ}{n+2β} \quad \text{ for } β\in\left[0, \tfrac{n}{2}\right) \text{ and } γ\in [0, n). $$ More precisely, we establish the global (in time) existence of small data solutions in the supercritical and critical regimes $p_1+p_2 \geq p_{\mathrm{Fuji}}\left(\frac{n+2 β}{2+γ}\right)$. In contrast, we prove finite-time blow-up of weak solutions, even for arbitrarily small initial data, in the subcritical range $2<p_1+p_2<p_{\mathrm{Fuji}}\left(\frac{n+2 β}{2+γ}\right)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23764 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A new critical exponent for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces Phan, Duc An Analysis of PDEs 35A01, 35B33, 35B44, 35L15, 35L71 In this paper, we investigate the critical exponent for a semi-linear damped wave equation involving a Hartree-type nonlinearity of the form $\mathcal{I}_γ\left(|u|^{p_1}\right)|u|^{p_2}, p_1, p_2>0, γ\in[0, n)$, with initial data taken in the homogeneous Besov spaces $\dot{B}_{2, \infty}^{-β}$, where $β\in\left[0, \frac{n}{2}\right)$. Our approach is based on deriving decay estimates for solutions to the associated linear damped wave equation with initial data belonging to $\dot{B}_{2, \infty}^{-β}$, combined with refined tools from Harmonic Analysis. As a consequence, we identify a new critical exponent given by $$ p_1+p_2:=p_{\mathrm{Fuji}}\left(\tfrac{n+2β}{2+γ}\right):=1+\tfrac{4+2γ}{n+2β} \quad \text{ for } β\in\left[0, \tfrac{n}{2}\right) \text{ and } γ\in [0, n). $$ More precisely, we establish the global (in time) existence of small data solutions in the supercritical and critical regimes $p_1+p_2 \geq p_{\mathrm{Fuji}}\left(\frac{n+2 β}{2+γ}\right)$. In contrast, we prove finite-time blow-up of weak solutions, even for arbitrarily small initial data, in the subcritical range $2<p_1+p_2<p_{\mathrm{Fuji}}\left(\frac{n+2 β}{2+γ}\right)$. |
| title | A new critical exponent for the semilinear damped wave equation with Hartree-type nonlinearity and initial data from homogeneous Besov spaces |
| topic | Analysis of PDEs 35A01, 35B33, 35B44, 35L15, 35L71 |
| url | https://arxiv.org/abs/2604.23764 |