This paper is devoted to study the existence of renormalized solutions of the parabolic Dirichlet equations of prototype: $\left\{\begin{array}{l}\frac{\mathrm{\partial }b(x,u)}{\mathrm{\partial }t}=Lu+f(x,t)\text{in}\mathrm{\Omega }\times (0,T),\\ Lu=\mathrm{d}\mathrm{i}\mathrm{v}(|\mathrm{\nabla }u{|}^{p\left(x\right)-2}\mathrm{\nabla }u+c(x,t)|u{|}^{\gamma \left(x\right)}+F)+l(x,t)|u{|}^{\delta \left(x\right)}\\ b(x,u){|}_{t=0}=b(x,u_0(x\left)\right)\text{in}\mathrm{\Omega },\\ u=0\text{on}\mathrm{\partial }\mathrm{\Omega }\times (0,T)\end{array}\right.$where $b(x,s),c(x,t),l(x,t),f(x,t)$ and the exponents of nonlinearities $p\left(x\right),\gamma \left(x\right),\delta \left(x\right)$ are given functions. The main contribution is the proof of an existence result without neither coercivity nor sign condition on non-linearities.

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具有非强制低阶项的非线性演化方程

本文致力于研究原型抛物型狄利克雷方程重整化解的存在性：$\left\{\begin{array}{l}\frac{\mathrm{\partial }b(X,你)}{\mathrm{\partial }吨}=\mathrm{大号}你+F(X,吨)\text{在}\mathrm{\Omega }\times (0,吨),\\ \mathrm{大号}你=\mathrm{d}\mathrm{一世}\mathrm{v}(|\mathrm{\nabla }你{|}^{p\left(X\right)-2}\mathrm{\nabla }你+C(X,吨)|你{|}^{\gamma \left(X\right)}+F)+l(X,吨)|你{|}^{\delta \left(X\right)}\\ b(X,你){|}_{吨=0}=b(X,{你}_0(X\left)\right)\text{在}\mathrm{\Omega },\\ 你=0\text{上}\mathrm{\partial }\mathrm{\Omega }\times (0,吨)\end{array}\right.$在哪里$b(X,s),C(X,吨),l(X,吨),F(X,吨)$和非线性指数$p\left(X\right),\gamma \left(X\right),\delta \left(X\right)$被赋予功能。主要贡献是证明存在结果既没有矫顽力也没有非线性的符号条件。