I have heard that people will be less spiritually inclined at the end of kaliyuga. The world will refurbish again to a utopian model of Satyayuga. Are there any references which states that Hinduism will exist throughout the period of Kaliyuga ?
Well, the Aitareya Brahmana has a beautiful way of expressing what the Yugas really mean. This is a very famous section in the story of Harishcandra.
Aitareya Brahmana 33.3.3:
कलिः शयानो भवति संजिहानस्तु द्वापरः ।
उत्तिष्ठंस्त्रेता भवति कृतं संपद्यते चरंश्चरैवेति ॥
Kali is when one sleeps, Dvapara is when he wakes up, Treta is when he stands up, and Krta is when he walks. Hence always walk (i.e. be dynamic).
चरन् वै मधु विन्दति चरन् स्वादुमुदुम्बरम् ।
सूर्यस्य पश्य श्रेमाणं यो न तन्द्रयते चरंश्चरैवेति ॥
Only by walking does one find honey, only by walking does one find sweet fruits. Look at the excellence of the Sun, who is always walking and never gets lazy or fatigued. Hence always walk (i.e. be dynamic).
So you see, Kaliyuga is really the state of ignorance and laziness, whereas Satya (or Krta) Yuga is really the state of knowledge and dynamism.
These verses are full of deep and meaningful symbolisms. 'Honey' and 'sweet fruits' can stand for either good results in this world or the highest spiritual achievement.
This is the true wise meaning of Yugas.
So to your question, we ourselves are responsible for bringing about a Kaliyuga or a Satyayuga. Hinduism will correspondingly decline or rise up again.
Bhagavan Krishna says about the span of His devotees being existent in this age of Kali, as 10,000 years.
𝐁𝐫𝐚𝐡𝐚𝐦𝐚𝐯𝐚𝐢𝐯𝐚𝐫𝐭𝐚 𝐏𝐮𝐫𝐚𝐧𝐚, 𝟒.𝟏𝟐𝟗. 𝟒𝟗 𝐭𝐨 𝟔𝟐
हे नाथ रमणश्रेष्ठ यासि गोलोकमुत्तमम्। अस्माकं का गतिश्चात्र भविष्यति कलौ युगे॥४९॥
Bhāgīrathi said- O lovable lord, you are now proceeding to Goloka. What shall be our position during the age of Kali?
कलेः पञ्चसहस्राणि वर्षाणि तिष्ठ भूतले। पापानि पापिनो यानि तुभ्यं दास्यन्ति स्नानत।।५०॥ मन्मन्त्रोपासकस्पर्शाद्धस्मीभूतानि तत्क्षणात्। भविष्यन्ति दर्शनाच्च स्नानादेव हि जाह्नवि।।५।।
Sri Krsna said- O Ganga, You remain on earth for five thousand years during Kaliyuga. All the sins that are washed in your water would disappear with the reciting of the mantras by the devotees and by the people who earn merit by having a look at you.
हरेर्नामानि यत्रैव पुराणानि भवन्ति हि। तत्र गत्वा सावधानमाभिः सार्धं च श्रोष्यसि॥५२॥
You would also listen to the reciting of the name of Hari and the reciting of the Purāņas.
पुराणश्रवणाच्चैव हरेर्नामानुकीर्तनात्। भस्मीभूतानि पापानि ब्रह्महत्यादिकानि च।।५३॥
With the reciting of the Purāṇas and the name of Hari, the sins like Brahmahatyā get destroyed in no time.
भस्मीभूतानि तान्येव वैष्णवालिङ्गनेन च। तृणानि शुष्ककाष्ठानि दहन्ति पावका यथा।।५४॥ तथाऽपि वैष्णवा लोके पापानि पापिनामपि। पृथिव्यां यानि तीर्थानि पुण्यान्यपि च जाह्नवि॥५५। मद्भक्तानां शरीरेषु सन्ति पूतेषु संततम्। मद्भक्तपादरजसा सद्यः पूता वसुंधरा।।५६॥
These sins are reduced to ashes with the touch of a Vaisnavas in the same way as the strew is burn it out in the fire flames or the dry-wood is also burnt out in fire. O Ganga, all the holy places which are located on earth, are all enshrined in the bodies of my devotees. The earth gets purified with the dust from the feet of my devotees.
सद्यः पूतानि तीर्थानि सद्यः पूतं जगत्तथा। मन्मन्त्रोपासका विप्रा ये मच्छिष्टभोजिनः॥५७।। मामेव नित्यं ध्यायन्ते ते मत्प्राणाधिकाः प्रियाः। तदुपस्पर्शमात्रेण पूतो वायुश्च पावकः॥५८॥
All the holy places get purified at once with the fall of the dust from the feet of my devotee. A Brāhmaṇa who is devoted to me and recites my mantras also consumes the naivedya and always remains devoted to me, he becomes dearer to me than my life. With the very touch of his body, the wind as well as the fire get purified.
कलेर्दशसहस्राणि मद्भक्ताः सन्ति भूतले। एकवर्णा भविष्यन्ति मद्भक्तेषु गतेषु च।।५९॥ मद्भक्तशून्या पृथिवी कलिग्रस्ता भविष्यति। एतस्मिन्नन्तरे तत्र कृष्णदेहाद्विनिर्गतः।।६०॥ चतुर्भुजश्च पुरुषः शतचन्द्रसमप्रभः। शङ्खचक्रगदापाधरः श्रीवत्सलाञ्छनः।।६।।
𝐃𝐮𝐫𝐢𝐧𝐠 𝐊𝐚𝐥𝐢𝐲𝐮𝐠𝐚 𝐦𝐲 𝐝𝐞𝐯𝐨𝐭𝐞𝐞𝐬 𝐰𝐨𝐮𝐥𝐝 𝐫𝐞𝐦𝐚𝐢𝐧 𝐨𝐧 𝐞𝐚𝐫𝐭𝐡 𝐟𝐨𝐫 𝐭𝐞𝐧 𝐭𝐡𝐨𝐮𝐬𝐚𝐧𝐝 𝐲𝐞𝐚𝐫𝐬. In the meantime the four-armed from, having the lustre of hundred of moons, illumining all the directions, holding sankha, cakra, gadā and padma, with Śrivatsa mark on his chest, emerged from the body of Krsna.
सुन्दरं रथामारुह्य क्षीरोदं स जगाम ह। सिन्धुकन्या च प्रययौ स्वयं मूर्तिमती सती॥६२॥
He at once mounted on the beautiful chariot and proceeded towards the ocean of milk. Lakşmi, the daughter of the ocean also transformed herself into a goddess and left the place.