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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.01877 |
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| _version_ | 1866910187389976576 |
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| author | Li, Shijun Li, Shujing Xu, Shaopeng |
| author_facet | Li, Shijun Li, Shujing Xu, Shaopeng |
| contents | In this paper, we consider the following nonlinear parabolic equation with non-coercive terms in \(R^N\) space \[ \dfrac{\partial u}{\partial t} -\nabla \cdot (a(x,t,u,\nabla u)+ Φ(x,t,\nabla u))=f, \text{ in }Ω\times (0,T). \] Here \(Ω\) is a bounded open set of \(R^N\) with the boundary \(\partial Ω\) satisfying Lipschitz condition. The Carathéodory function \(Φ\) is restricted by $|Φ(x,t,s)|\le c(x,t)|s|^γ$ with parameters depending on $p$ and $N$. And the initial value $u(x,0)=u_0(x)$. For convenience, we define the domain $Q := Ω\times (0,T)$ and the boundary similarly. Then for $f\in L^1(Q)$ and $u_0\in L^1(Ω)$, we prove the existence and uniqueness of a renormalized solution via truncation methods, monotone operator theory, and a prior gradient estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01877 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Renormalized Solution for the Nonlinear Parabolic Problem with Lower Order Terms Li, Shijun Li, Shujing Xu, Shaopeng Analysis of PDEs In this paper, we consider the following nonlinear parabolic equation with non-coercive terms in \(R^N\) space \[ \dfrac{\partial u}{\partial t} -\nabla \cdot (a(x,t,u,\nabla u)+ Φ(x,t,\nabla u))=f, \text{ in }Ω\times (0,T). \] Here \(Ω\) is a bounded open set of \(R^N\) with the boundary \(\partial Ω\) satisfying Lipschitz condition. The Carathéodory function \(Φ\) is restricted by $|Φ(x,t,s)|\le c(x,t)|s|^γ$ with parameters depending on $p$ and $N$. And the initial value $u(x,0)=u_0(x)$. For convenience, we define the domain $Q := Ω\times (0,T)$ and the boundary similarly. Then for $f\in L^1(Q)$ and $u_0\in L^1(Ω)$, we prove the existence and uniqueness of a renormalized solution via truncation methods, monotone operator theory, and a prior gradient estimates. |
| title | Renormalized Solution for the Nonlinear Parabolic Problem with Lower Order Terms |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.01877 |