Well-posedness and kernel stability for diffusion equations with mixed measure-valued memory
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908985322373120 |
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| author | Ishizaka, Hiroki |
| author_facet | Ishizaka, Hiroki |
| contents | We investigate a linear diffusion equation incorporating historical effects, characterised by a finite non-negative Borel measure on \((0, \mathfrak T]\). This approach accommodates both distributed memory and discrete delays within a unified weak formulation. The measure-valued framework encompasses the memory-free scenario, absolutely continuous kernels, purely atomic delay kernels, and mixed regimes. Our principal result is a finite-time well-posedness theorem for arbitrary finite measures, including kernels with atomic components. More precisely, we prove existence and uniqueness of weak solutions on \((-τ_{\max},\mathfrak T]\) and derive stability bounds with constants depending explicitly on \(\mathfrak T\), \(μ((0,\mathfrak T])\), and the coercivity and boundedness parameters of the bilinear forms. Subsequently, we demonstrate continuous dependence on the kernel over fixed time intervals, leading to regime-consistency results such as vanishing-memory limits and concentration to a discrete delay. For a restricted dissipative subclass of absolutely continuous kernels, we identify a positive-type condition that results in an energy inequality, and we provide verifiable sufficient criteria, including complete monotonicity, along with an internal-variable representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19099 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Well-posedness and kernel stability for diffusion equations with mixed measure-valued memory Ishizaka, Hiroki Analysis of PDEs We investigate a linear diffusion equation incorporating historical effects, characterised by a finite non-negative Borel measure on \((0, \mathfrak T]\). This approach accommodates both distributed memory and discrete delays within a unified weak formulation. The measure-valued framework encompasses the memory-free scenario, absolutely continuous kernels, purely atomic delay kernels, and mixed regimes. Our principal result is a finite-time well-posedness theorem for arbitrary finite measures, including kernels with atomic components. More precisely, we prove existence and uniqueness of weak solutions on \((-τ_{\max},\mathfrak T]\) and derive stability bounds with constants depending explicitly on \(\mathfrak T\), \(μ((0,\mathfrak T])\), and the coercivity and boundedness parameters of the bilinear forms. Subsequently, we demonstrate continuous dependence on the kernel over fixed time intervals, leading to regime-consistency results such as vanishing-memory limits and concentration to a discrete delay. For a restricted dissipative subclass of absolutely continuous kernels, we identify a positive-type condition that results in an energy inequality, and we provide verifiable sufficient criteria, including complete monotonicity, along with an internal-variable representation. |
| title | Well-posedness and kernel stability for diffusion equations with mixed measure-valued memory |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.19099 |