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In this article, we study the following noncoercive quasilinear parabolic problem ∂u∂t−diva(x,t,u,∇u)+ν∣u∣s−1u=λ∣u∣p−2u∣x∣p+finQT,u=0onΣT,u(x,0)=u0inΩ,\left\{\begin{array}{ll}\frac{\partial u}{\partial t}-\hspace{0.1em}\text{div}\hspace{0.1em}a\left(x,t,u,\nabla u)+\nu {| u| }^{s-1}u=\lambda \frac{{| u| }^{p-2}u}{{| x| }^{p}}+f& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{Q}_{T},\\ u=0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{\Sigma }_{T},\\ u\left(x,0)={u}_{0}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\end{array}\right. with f∈L1(QT)f\in {L}^{1}\left({Q}_{T}) and u0∈L1(Ω){u}_{0}\in {L}^{1}\left(\Omega ) and show the existence of entropy solutions for this noncoercive parabolic problem with Hardy potential and L1-data.

This paper is devoted to study the global existence of solutions of the hyperbolic Dirichlet equation u t ⁢ t = L ⁢ u + f ⁢ ( x , t ) in ⁢ Ω T = Ω × ( 0 , T ) , u_{tt}=Lu+f(x,t)\quad\text{in }\Omega_{T}=\Omega\times(0,T), where L is a nonlinear operator and ϕ ⁢ ( x , t , ⋅ ) {\phi(x,t,\cdot\,)} , f ⁢ ( x , t ) {f(x,t)} and the exponents of the nonlinearities p ⁢ ( x , t ) {p(x,t)} and μ ⁢ ( x , t ) {\mu(x,t)} are given functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove existence result of renormalized solutions in the setting of Musielak-Orlicz spaces W10 Lφ(Ω) for the following strongly nonlinear Dirichlet problem A(u) + g(x, u,∇u) = f in Ω, where A is a Leray-Lions operator acting from its domain D(A) ⊂ W10 Lφ(Ω) into its dual, while g(x, u,∇u) is a nonlinear term having a growth conditions with respect only to ∇u, and does not satisfy any sign condition. The right-hand side f belongs to L1(Ω). A modular-inequality of Poincaré type in this setting is also proved (see Lemma 2.5). [ABSTRACT FROM AUTHOR]